PDF《Chapter 2》

Chapter

2 Building LP-Risk Models of LP-Modeling Class It may be a surprise for some mathematicians that the problem data ? the model, explaining the data should be considered as the basic one for any area of science.

Kalman Risk models of the LP-modeling class are fully described in the works concerning technical applications by I. Ryabinin and A. Mozhaev. We have already described the LP-model of this class for solving dif?cult economic problems. The procedure of building the LP-model of this class is the following: development of the risk scenario, writing down the L-risk model according to the scenario, transition from the L-risk model to the P-risk model. The model of the LP-modeling class is used as the basis for building risk model of classes LP-classi?cation, LP-ef?ciency and LP-forecasting, i.e. for the identi?- cation of the model of these classes by statistical data. Therefore the description of the issues concerning building the model of LP-modeling class is of greatest impor- tance. The Swiss mathematician and the author of the so called Kalman ?lter deter- mined the requirements to a model in science [30]. He wrote that it may be a sur- prise for some mathematicians that the problem data ? the model, explaining the data should be considered as the basic one for any area of science. It has deep math- ematical content and close links with Kolmogorov'

s theory of probabilities. We believe that another important requirement to a mathematical model is that of the possibility of detailed and transparent analysis of the model, results and data for management purposes. Neither scoring techniques, nor neural networks meet this requirement. The uniqueness principle places an emphasis on the undeniable fact that science results should be obtained from objective data analysis, and not from random self-assertive manipulations with model. The LP-model of LP-modeling class conform to Kalman'

s rule, as well as the LP-model of derivative classes LP-classi?cation, LP-ef?ciency and LP-forecasting. The independent use of the LP-model of LP-modeling class has been considered for the following tasks: E.D. Solozhentsev, Risk Management Technologies, Topics in Safety, Risk, Reliability and Quality 20, DOI 10.1007/978-94-007-4288-8_2, ? Springer Science+Business Media Dordrecht

2012 29

30 2 Building LP-Risk Models of LP-Modeling Class ? I3-technologies of solving dif?cult problems (Sect. 1.1, Chaps.

10 and 21);

? LP-management of a transport company ef?ciency (Chap. 11);

? the LP-model of the company management failure risk (Chap. 15);

? the LP-model of a bank'

s operation risk (Chap. 16);

? LP-analysis and management of the processes non-validity risk (Chap. 18);

? the LP-model of ?re-hazardous objects insurance risk (Chap. 20). 2.1 Perfect Disjunctive Normal Form In economy and engineering the possible system states (complete set) can be always written down as a perfect disjunctive normal form (PDNF) taking into account the two states of each event-parameter (in engineering) [57] or taking into account GIE for each event-parameter (in economy) [85]. The total number of different system events-states is determined by expression (1.7). All these states can be written down in matrix form. The state appearance probability in statistical data is calculated by probabilities of events Z1,Z2,...,Zn (to be more precise―their corresponding gra- dations) appearing in statistical data. Orthogonality of system states in a KB. L-function for all possible system states is given by Y = Y1 ∨ Y2 ∨ ・・・ ∨ Yk ∨ ・・・ ∨ YN , (2.1) where the state is determined by the L-function with all L-variables: Yk = Z1 ∧ ・・・ ∧ Zj ∧ ・・・ ∧ Zn. (2.2) Each L-derivative takes a lot of values according to the number of gradations or intervals, into which the parameter is divided. L-functions for two different states (objects), for example Yk = Z1 ∧ ・・・ ∧ Zjr ∧ ・・・ ∧ Zn;

Yk+1 = Z1 ∧ ・・・ ∧ Zjr+1 ∧ ・・・ ∧ Zn, (2.3) are orthogonal, as Zjr and Zjr+1 belong to the same GIE: Zjr ∧ Zjr+1 = 0. The orthogonality property of the addends of the system states risk L-function allows passing over from L-functions to algebraic expressions for probabilities (risk), anal- ysis the state risk according to the contribution of events-gradations, calculating transition probabilities, as well as overcoming exponential computational complex- ity of the algorithm. 2.2 The Shortest Paths of Successful Operation Building the LP-risk model on the shortest paths of successful operation (SPSO) is widely spread in engineering [57], when electric, water gas or any other scheme of a 2.3 Minimal Failure Cross-Sections

31 Fig. 2.1 Structural model of the bridge system, device, etc. exists. In economy the LP-model of the system state failure risk is built according to the risk scenario or the failure risk graph model, which connect elements Z1,...,Zn. The L-function of risk is written down as minimal failure cross-sections or the shortest paths of successful operation [51, 57]. However, now one needs orthogo- nalization of L-functions in order to obtain P-function of risk, but this procedure is not a real problem when one has special Software and modern computers. Example Electric circuit of the bridge type (Fig. 2.1) will be written down in disjunctive normal form (DNF) as a logical sum of the shortest paths of successful operation [57]: Y = Z1Z3 ∨ Z2Z4 ∨ Z1Z5Z4 ∨ Z2Z5Z3. (2.4) After orthogonalization (2.4) we obtain P-model of risk: Pi = p2p4 + p1p3 + q1p2p3q4p5 + p1q2q3p4p5 ? p1p2p3p4. (2.5) 2.3 Minimal Failure Cross-Sections It does not matter if we write down the L-function for success or failure, as the failure probability q =

1 ? p, where p is success probability. It is often important to analyze failure risk itself. Then it is more convenient to write down instead (2.4) the system failure L-function as minimal cross-sections of elements failure [57] Y = Z1Z2 ∨ Z3Z4 ∨ Z1Z5Z3 ∨ Z3Z5Z4 (2.6) and then perform orthogonalization of this functions and write down the P- polynomial of risk. 2.4 Associative LP-Risk Models The system states failure risk scenario can be associative [85]. For example, a failure causes one, two, ... or all initiating events from Z1,Z2,...,Zn. L-failure risk model

32 2 Building LP-Risk Models of LP-Modeling Class is written down according to this scenario (which is a PDNF subset). Here we also need orthogonalization of L-functions in order to obtain P-function of risk. Example L-function of the associative model failure risk: Y = Z1 ∨ Z2 ∨ ・・・ ∨ Zj ∨ ・・・ ∨ Zn, (2.7) where Z1,...,Zn are logical variables for the state parameters. Logical function of failure risk after the orthogonalization of the associative risk model (2.7): Y = Z1 ∨ Z2Z1 ∨ Z3Z1Z2 ∨ ・・・. (2.8) From (2.8) P-function of the associative model failure risk: P(Y1 = 0) = P1 + P2 ・ (1 ? P1) + P3 ・ (1 ? P2) ・ (1 ? P1)2.9) where Pj = P{Zj } is the probability of independent events Zj leading to failure Y1. A limited set of events. As A. A. Losev demonstrated PDNF allows building associative the LP-model for a limited set of events [85], For example, when one or any two events occur L-risk model will be written down as Y1 = Z1Z2Z3Z4 ∨ Z2Z1Z3Z4 ∨ Z3Z1Z2Z4 ∨ Z4Z1Z2Z3 ∨ Z1Z2Z3Z4 ∨ Z1Z3Z2Z4 ∨ Z1Z4Z2Z3 ∨ Z2Z3Z1Z4 ∨ Z2Z4Z1Z3 ∨ Z3Z4Z1Z2. (2.10) In this L-model all L-items are mutually orthogonal, which allows writing failure risk P-model (polynomial) at once on the assumption of the independence of events Z1, Z2, Z3, Z4: P(Y1 = 0) = p1q2q3q4 + p2q1q3q4 + p3q1q2q4 + p4q1q2q3 + p1p2q3q4 + p1p3q2q4 + p1p4q2q3 + p2p3q1q4 + p2p4q1q3 + p3p4q1q2. (2.11) 2.5 Tabular Form of t........