PART V. Practices and Emerging Applications
5.3 Combinatorial Reliability Optimization Problems of a
This section considers problems having non- series structures including mixed series–parallel structures and general structures.
5.3.1 Mixed Series–Parallel System Optimization Problems
As mentioned earlier, a mixed series–parallel structure is defined as one in which the relation between any two stages is in series or parallel. For such a structure, the system reliability function is not expressed as a product form of the reliabilities of all stages, but as a multinomial form of the reliabilities of multiple paths. Therefore, the system reliability depends on its system structure, and so is very complicated to compute. However, it is known that the system reliability of a mixed series–parallel structure can be figured out on the order of a polynomial function of the number of stages, so that the complexity of computing the
Combinatorial Reliability Optimization 103 system reliability does not increase exponentially
as the number of stages in the system increases.
One often sees mixed series–parallel structures in international telephone systems. For example, each international telephone system between any two countries is composed of two major subsystems. One of them is a wire system, and the other one is a wireless system. In the whole system, all calls from one country to another country are collected together at an international gateway; some of the calls are transmitted through the wireless system from an earth station of one country to the corresponding earth station of the other country via a communications satellite; the rest of the calls are transmitted through the wire system (being composed of terminals and optical cables) from the international gateway to the corresponding international gateway of the other country. For these international communications, both the wire and the wireless systems are equipped with multiple modules in parallel, each being composed of various chipsets so as to increase the whole system reliability. Thus, the associated design issue of how many modules to be installed multiply in each subsystem for an optimal international telephone system can be handled as a mixed series–parallel system optimization problem.
Only a few researches have considered the re- liability optimization issue for such mixed series–
parallel systems. For example, Burton and Howard [27] have dealt with the problem of maximizing system reliability subject to one resource con- straint such as budget restriction. Their algo- rithm is based on a basic dynamic programming.
Recently, Cho [28] has proposed a dominance- sequence-based dynamic programming algorithm for the problem with binary variables and an- alyzed its computational complexity, and then compared it with the algorithm of Burton and Howard [27].
We now introduce the dominance-sequence- based dynamic programming algorithm of Cho [28]. The problem, dealt with by Cho [28], is one with a mixed series–parallel system structure. For developing the algorithm, the following assumptions are made. First, the
resource consumption for the entire system is represented by a discrete linear function. Second, the resource coefficients are all integer valued. The proposed algorithm is composed of two phases.
The first phase is to construct a reduction-order graph, which is defined as a directed graph having the precedence relation between nodes in the proposed mixed series–parallel system. Referring to Satyanarayana and Wood [2], the mixed series–
parallel system can be reduced to a one-node network in the complexity order of O(|E|), where |E| denotes the cardinality of the set of edges in the given mixed series–parallel system.
Two examples of the reduction-order graphs are depicted in Figure 5.1. The order of reducing the given graph is not unique, but the structure of the reduction-order graph remains the same as that of the given graph, which is represented as a tree.
The number of end nodes of the graph depends on the given system structure. The second phase is concerned with a process of reducing two stages (including series-stage reduction and parallel-stage reduction) into one as follows.
Series-stage reduction. Suppose that stagesi and k are merged together in series relation and to generate a new stages. Then, it can be processed asrijrkl→rs,j×l andcijm+cklm→cs,jm×lfor allm. Accordingly, the variableys,j×lwill be included in the stageswith its reliability and consumption of resourcematrs,j×landcs,jm×l, respectively.
Parallel-stage reduction. Suppose that stagesiand k are also merged together in parallel relation and to generate a new stage s. Then, it can be processed asrij+rkl−rijrkl→rs,j×l and cijm+ cmkl→cms,j×l. Accordingly, the variablexs,j×l will be newly included in the stageswith its reliability and consumption of resource m at rs,j×l and cms,j×l, respectively.
The above discussions on stage reduction are now put together to formulate the dominance- sequence-based dynamic programming proce- dure:
Step 0. (Ordering) Apply the algorithm of Satya- narayana and Wood [2] to the proposed mixed series–parallel system and find the
104 System Reliability and Optimization
Figure 5.1. Two examples of the reduction-order graph
reduction-order graph of the system. Go to Step 1.
Step 1. (Initialization) Arrange all stages accord- ing to the reduction-order graph, and set σ= ∅and σ¯ = {(1), (2), . . . , (|I|)},which represents the set of the reduced stages and non-reduced ones respectively. Go to Step 2.
Step 2. (Reduction) If any two selected stages are in series relation, then perform the series-stage reduction procedure for the two stages. Otherwise, perform the parallel- stage reduction procedure. Go to Step 3.
Step 3. (Termination) As σ¯ becomes a null set, terminate the procedure. Otherwise, go to Step 2.
To illustrate the dominance-sequence-based dynamic programming procedure, a numerical example is solved. The problem data are listed in Table 5.7. And the structure of Figure 5.1(a) is used as representing the example.
At the first iteration, the series-stage reduction procedure is applied to merge stages b and c into stage 3. The variables of stage 3 are generated in Table 5.8.
At the second iteration, the parallel-stage reduction procedure is applied to merge stages 3 and d into stage 2. At the last iteration, the series- stage reduction procedure is applied to merge stages 2 and a into stage 1, whose variables are given in Table 5.9.
The first and second variables are not feasible because the weight consumptions are too great.
Thus, the optimal system reliability is found at 0.9761, and its cost and weight consumptions are found at 28 and 15 respectively.
Now, the computational complexity of the dominance-sequence-based dynamic program- ming algorithm is analyzed and compared with the basic dynamic programming algorithm of Bur- ton and Howard [27]. For the one-dimensional problem, the following properties are satisfied.
Proposition 1.(Cho [28]) If the reduction-order graph of the given system is a tree with only one end node, then the computational complexity of the dominance-sequence-based dynamic program- ming algorithm of Cho [28] is in the order of O(|I|J C),where|I|and|Ji|denote the cardinal- ities ofIandJirespectively;Jirepresents the set of variables used at stagei, andJ=maxi∈I{|Ji|}.
Proposition 2.(Cho [28]) If the reduction-order graph of the given system is a tree with more than one end node, then the computational complex- ity of the dominance-sequence-based dynamic pro- gramming algorithm is in the order ofO(|I|C2).
Proposition 3.(Cho [28]) The computational complexity of the basic dynamic programming algorithm of Burton and Howard [27] is in the order of O(|I|C2) for any mixed series–parallel network.
Combinatorial Reliability Optimization 105
Table 5.7. Problem data (reliability, cost, weight) for the dominance-sequence- based dynamic programming algorithma
Component Stage a Stage b Stage c Stage d
type
1 0.92,7,5 0.90,3,9 0.92,5,11 0.80,3,3 2 0.98,8,3 0.98,11,4 0.90,5,6
aAvailable cost and weight are 30 and 17, respectively.
Table 5.8. Variables generated at the first iteration of the algorithm of Cho [28]
j r3j(=rbj×rcj) c3j(=cbj×ccj) w3j(=wbj×wcj) 1 0.828(=0.9×0.92)a 8(=3+5) 20(=9+11)
2 0.882 14 13
3 0.9016 13 14
4 0.9604 19 7
aThe variable is fathomed because its weight consumption is not feasible (i.e.20>17).
Table 5.9. Variables generated at the third iteration of the algorithm of Cho [28]
j r1j(=r2j×raj) c1j(=c2j×caj) w1j(=w2j×waj) 1 0.9569(=0.9764×0.98)a 21(=17+4) 18(=16+2)
2 0.9607a 20 19
3 0.9723 26 12
4 0.9761 28 15
aThe variable is fathomed due to resource violation.
As seen in the computational complexity anal- ysis of the one-dimensional case, the dominance- sequence-based dynamic programming algorithm may depend on the system structure, whereas the basic dynamic programming algorithm of Bur- ton and Howard [27] does not. Therefore, the dominance-sequence-based dynamic program- ming algorithm may require a reduced computa- tional complexity for the systems of a tree struc- ture with one end node. For the detailed proof, refer to Burton and Howard [27] and Cho [28].
For multi-dimensional problems, the computa- tional complexity is characterized below.
Proposition 4.(Cho [28]) The computational complexity of the dominance-sequence-based dy- namic programming algorithm of Cho [28] is in the
order ofO(|M|J|I|)for any mixed series–parallel network.
Proposition 5.(Cho [28]) The computational complexity of the basic dynamic programming algorithm of Burton and Howard [27] is in the order ofO(|I|C2|M|)for any mixed series–parallel network.
In general, the number of constraints is smaller than the number of stages, so that the computational complexity of the proposed algorithm is larger than that of Burton and Howard [27], even in the situation whereJC2. There exist a number of variables having not the same resource consumption rate but the same reliability, so that the number of the variables generated is not bounded by the valueC at each
106 System Reliability and Optimization
iteration of the algorithm. For the detailed proof, refer to Burton and Howard [27] and Cho [28].
5.3.2 General System Optimization Problems
One often sees general-structure systems in the area of telecommunications. Telecommunication systems are composed of many switches, which are interconnected with one another to form complex mesh types of network, where each switch is equipped with multiple modules in parallel, each being composed of many chipsets to process traffic transmission operations so as to increase the whole switch reliability. Thus, the associated design issue of how many modules to be installed multiply in each switch for an optimal telecommunication system can be handled as a general system reliability optimization problem.
For the reliability optimization problems of complex system structures, no efficient optimal solution method has yet been derived, and only a few heuristic methods [29, 30] have been proposed. The heuristic methods, however, have been applied to small-sized problems, because, as stated earlier, the computational complexity of the system reliability of complex structures may increase exponentially as the number of stages in the system increases.
Aggarwal [29] proposed a simple heuristic algorithm for a reliability maximization problem (NRP) to select a stage having the largest ratio of the relative increment in reliability to the increment in resource usage and to add a redundant unit at the stage for increasing redundancy. That is, a redundant unit is added to the stage where its addition has the largest value of the selection factorFi(yi)defined as follows:
Fi(yi)= Qs(yi) .
m∈Mgim(yi)
whereQs(yi)represents increment in reliability when a redundant unit is added to stageihaving yiredundant units such that
Qs(yi)=Qs(Q1, . . . , Qi, . . . , Q|I|)
−Qs(Qˆ1, . . . ,Qˆi, . . . ,Qˆ|I|)
1 2
3 4
5
Figure 5.2. A bridge system
whereQs(Q1, . . . , Q|I|)andQi=(1−ri)yirep- resent the failure probabilities of the system and stage i, respectively, and Qj= ˆQj ∀j=i, and Qˆi=(1−ri)yi+1. For the detailed step-by-step procedure, refer to Aggarwal [29].
In order to illustrate the algorithm of Aggarwal [29] for the bridge system structure given in Figure 5.2, a numerical example is solved with the data of Table 5.10; the results are presented in Table 5.11.
Shi [30] has also proposed a greedy-type heuristic method for a reliability maximization problem (NRP), which is composed of three phases. The first phase is to choose the minimal path with the highest sensitivity factoralfrom all minimal paths of the system as
al=
.
i∈lRi(yi)
i∈Pl
m∈M[gmi (yi)/|M|Cm] forl∈L which represents the ratio of the minimal path re- liability to the percentage of consumed resources by the minimal path. The second phase is to find within the chosen minimal path the stage having the highest selection factorbiwhere
bi= R−i(yi)
m∈M[gmi (yi)/|M|Cm] fori∈Pl Table 5.10. Problem data for the bridge system problem
Stage 1 2 3 4 5
ri 0.7 0.85 0.75 0.8 0.9
ci(=cost) 3 4 2 3 2
Constraints:C≤19
Combinatorial Reliability Optimization 107
Table 5.11. Illustration of the stepwise iteration of the algorithm of Aggarwal [29]
Iteration Stage Fi(yi)
ci Qs(%)
1 2 3 4 5 1 2 3 4 5
1 1 1 1 1 1 1.77 1.34 2.76a 1.07 0.27 14 10.9 2 1 1 2 1 1 0.53 0.62 0.69 1.15a 0.24 16 5.4
3 1 1 2 2 1 — — — — — 19 2.6
aThe stage to which a redundant unit is to be added.
Table 5.12. Illustration of the stepwise iteration of the algorithm of Shi [30]
Stage al bi
1 2 3 4 5
ci P1 P2 P3 P4 1 2 3 4 5
1 1 1 1 1 14 1.62 2.28a 1.20 1.36 — — 7.1b 5.1 — 1 1 2 1 1 16 1.62 2.04 1.20 1.36 — — 4.5 5.1b —
1 1 2 2 1 19 — — — — — — — — —
aThe minimal path set having the largest sensitivity factor.
bThe stage to which a redundant unit is to be added.
The second phase checks feasibility and, if feasi- ble, allocates a component to the selected stage.
For the detailed step-by-step procedure, refer to Shi [30].
In order to illustrate the algorithm of Shi [30], a numerical example is solved with the data of Table 5.10, and Table 5.12 shows the stepwise results to obtain the optimal solution, which also shows the minimal path sets P1= {1,2}, P2= {3,4},P3= {1,4,5},P4= {2,3,5}.
As seen in the above illustration, the optimal solution obtained from the algorithm of Shi [30] is the same as that of the algorithm of Aggarwal [29].
Recently, Raviet al.[31] proposed a simulated- annealing technique for the reliability optimiza- tion problem of a general-system structure having the objective of maximizing the system reliability subject to multiple resource constraints.