CHAPTER 11
Optimal Redundancy in Multistate Systems
Solving the problems of optimal redundancy allocation for multistate systems (MSS) consisting of multistate units is more laborious than solving analogous problems for systems that have only two states: normal operation and failure.
Today, this problem is investigated in detail. To begin with, the works of Gregory Levitin and Anatoly Lisnjanskij have to be mentioned (a complete list of their papers is presented in the Bibliography to the chapter).
To gain a clearer perspective of this type of problem, we begin with a simple numerical example. Consider a series system of two different multistate units, each of which is characterized by several levels of performance. Performance may be measured by various physical values. Effectiveness of such system operation depends on the levels of performance of Unit 1 and Unit 2. These units are characterized by the parameters presented in Table 11.1 and Table 11.2.
TABLE 11.1 Characterization of Unit 1
| Level of performance (W1) | Probability p1 | Cost of a single unit |
|---|---|---|
| 100% | p11 = Pr{W1 = 100%} = 0.9 | c1 = 1 |
| 70% | p12 = Pr{W1 = 100%} = 0.05 | |
| 40% | p13 = Pr{W1 = 100%} = 0.04 | |
| 0% | p14 = Pr{W1 = 100%} = 0.01 |
TABLE 11.2 Characterization of Unit 2
| Level of performance (W1) | Probability p2 | Cost of a single unit |
|---|---|---|
| 100% | p21 = Pr{W2 = 100%} = 0.8 | c2 = 2 |
| 80% | p22 = Pr{W2 = 80%} = 0.18 | |
| 20% | p23 = Pr{W2 = 20%} = 0.01 | |
| 0% | p24 = Pr{W2 = 0%} = 0.01 |
Assume that the performance effectiveness of each unit can be improved by using loaded ...
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