RELIABILITY AND RISK OPTIMIZATION OF MULTISTATE SYSTEMS WITH APPLICATION TO PORT TRANSPORTATION SYSTEM

The complexity of technical systems’ operation processes and its influence on the changing in time systems’ structures and their components’ reliability parameters posses a difficulty to first meet in real and then to fix and analyse those structures and reliability parameters. By constructing a joint model of reliability of complex technical systems at variable operation conditions, which links a semimarkov modelling of system operation processes with multi-state approach to system reliability analysis, we find the system’s main reliability characteristics. Consequently, we use linear programming to build a model of complex technical systems reliability optimization. We investigate the model’s application in marine transport, specifically in reliability and risk optimization of a bulk cargo transportation system. The tools we develop can be used in reliability evaluation and optimization of a very wide class of real technical systems operating at varying conditions that influence their reliability structures and the reliability parameters of their components. Consequently, the tools we developed can be implemented by reliability practitioners from both maritime transport industry and other industrial sectors.


NOMENCLATURE
), (t Z i z is a system operation process; a system operation state; , ,..., 2 , p is an optimal transient probability; U is a particular reliability state of the system; z u ,..., 2 , 1 = ; ) (u T is a lifetime of a system in the reliability state subset } ,..., 1 , is a conditional reliability function of a system at the operational state b z ; is an optimal unconditional reliability function of a system; ) (u μ is a mean lifetime of the system in the reliability state subset } ,..., 1 , is a mean lifetime of the system in the reliability particular state u; R is a critical state of the multi-state system; r (t) is an optimal risk function of the multi-state system.

INTRODUCTION
Most real technical systems are very complex because they are composed of large numbers of components and subsystems and have high operating complexity. The complexity of the systems' operation processes and its influence on the changing in time systems' structures and their components' reliability parameters posses a difficulty to first meet in real and then to fix and analyse those structures and reliability parameters. A convenient tool to investigate this problem is a semi-markov [2] modelling of the system operation process linked with a multi-state approach for the system reliability analysis [1,4,[9][10] and a linear programming for the system reliability optimization [3]. Using this approach, it is possible to find this complex system's main reliability characteristics including the system reliability function, the system mean lifetimes in the reliability states subsets and the system risk function [4,6,8]. Having those characteristics it is possible to optimize the system operation process to get optimal values [8]. To this end the linear programming [3] can be applied to maximize the mean value of the system lifetime in the subset of the system reliability states, which are not worse than the system critical reliability state.

SYSTEM RELIABILITY AT VARIABLE OPERATIONS PROCESS
We suppose that the system has v different operation states during its operation process. Thus, we can define the system operation process ), (t Z , , 0 > +∞ ∈< t as the process with discrete operation states from the set }.
is the conditional reliability function while the system is at the operational state , b z b = 1,2,…,v . Next, we denote the system lifetime in the reliability state is the conditional reliability function of the system while the system is at the operational state . b z In the case when the system operation time is large enough, the unconditional reliability function of the system is given by The mean values of the system lifetimes in the reliability and the mean values of the system lifetimes in the particular reliability state , u are [4] ), that the system is in the subset of reliability states worse than the critical state r, r ∈ {1,...,z} while it was in the state z at the moment t = 0 is called a risk function of the multi-state system [4].
Under this definition, from (3), we have (6) and if τ is the moment when the risk exceeds a permitted level δ , then if it exists, is the inverse function of the risk function r(t).

OPTIMAL TRANSIENT PROBABILITIES MAXIMIZING SYSTEM LIFETIME
Considering the equation (3), it is natural to assume that the system operation process has a significant influence on the system reliability. This influence is also clearly expressed in the equation (4) for a fixed } ,..., 2 , 1 { z r ∈ and with the following bound constraints are fixed mean values of the system conditional lifetimes in the reliability state subset } ,..., are respectively the lower and upper bounds of the unknown transient probabilities b p . Now, we can obtain the optimal solution to the formulated by (8)-(10) the linear programming problem, i.e. we can find the optimal values b p of the transient probabilities , b p b = 1,2,…, ν , that maximize the objective function given by (8). First, we arrange the system conditional lifetime mean values ), are fixed mean values of the system conditional lifetimes in the reliability state subset } ,..., are the lower and upper bounds of the unknown probabilities and we fix the optimal solution that maximize (12) in the following way: Finally, after making the inverse to (11) substitution, we get the optimal limit transient probabilities From the above, replacing r by , u , ,..., 2 , 1 z u = we obtain the corresponding optimal solutions for the mean values of the system unconditional lifetimes in the reliability state subsets } ,..., Further, according to (3), the corresponding optimal unconditional multistate reliability function of the system is ) , where ) , and by (5) the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are of the form Moreover, considering (6) and (7), the corresponding optimal system risk function and the moment when the risk exceeds a permitted level δ , respectively are given by

OPTIMAL SOJOURN TIMES OF COMPLEX TECHNICAL SYSTEM OPERATION PROCESS
Replacing the limit transient probabilities b p of the system operation process at the operation states by their optimal values b p and the mean values b M of the unconditional sojourn times at the operation states by their corresponding unknown optimal values b M maximizing the mean value of the system lifetime in the reliability states subset } ,..., 1 , { z r r + , we get the system of equations After simple transformations the above system takes the form M are unknown variables we want to find, b p are optimal transient probabilities and b π are steady probabilities.
Since the system of equations is homogeneous and it can be proved that the determinant of its main matrix is equal to zero, then it has nonzero solutions and moreover, these solutions are ambiguous. Thus, if we fix some of the optimal values b M of the mean values b M of the unconditional sojourn times at the operation states, for instance by arbitrary fixing either one or multiple of them, we may find the values of the remaining ones and using this method arrive at the solution of this equation.
Another very useful and much easier applicable in practice tool that can help in planning the operation processes of complex technical systems are the system operation process optimal mean values of the total system operation process sojourn times b θ at the particular during the fixed system operation time θ . They can be obtained by replacing the transient probabilities b p at the operation states b z with their optimal values b p . This results in the following expession The knowledge of the optimal values b M of the mean values of the unconditional sojourn times and the optimal mean values b M ˆ of the total sojourn times at the particular operation states during the fixed system operation time may be the basis for changing the complex technical systems operation processes in order to ensure that these systems operate both more reliably and more safely. This knowledge may also be useful in these systems operation cost analysis.

THE BULK CARGO TRANSPORTATION SYSTEM RELIABILITY AND RISK
The considered bulk cargo terminal placed at the Baltic seaside is designated for storage and reloading of bulk cargo, but its primary activity is loading bulk cargo on board the ships for export. There are two independent transportation systems: the system of reloading rail wagons and the system of loading vessels.
Cargo is brought to the terminal by trains consisting of self-discharging wagons, which are discharged to a hopper and then by means of conveyors transported into one of four storage tanks (silos). Loading of fertilizers from storage tanks on board the ship is done by means of special reloading system which consists of several belt conveyors and one bucket conveyor which allows the transfer of bulk cargo in a vertical direction. Researched system is a system of belt conveyors, referred to as the transport system.
In the conveyor reloading system we distinguish three bulk cargo transportation subsystems, the belt conveyors S 1 , S 2 and S 3 . The conveyor loading system is composed of six bulk cargo transportation subsystems, the dosage conveyor S 4 , the horizontal conveyor S 5 , the horizontal conveyor S 6 , the sloping conveyors S 7 , the dosage conveyor with buffer S 8 , the loading system S 9 .
The bulk cargo transportation subsystems are built, respectively: -the subsystem Taking into account the operation process of the considered system we distinguish the following as its three operation states: -an operation state − 1 z loading fertilizers from rail wagons on board the ship is done using S 1 , S 2 , S 3 , S 6 , S 7 , S 8 and S 9 subsystems.
-an operation state − 2 z discharging rail wagons to storage tanks or hall when subsystems S 1 , S 2 and S 3 , are used, -an operation state − 3 z loading fertilizers from storage tanks or hall on board the ship is done by using S 4 , S 5 , S 6 , S 7 , S 8 and S 9 , subsystems.
The limit values of the bulk cargo transportation systems operation process transient probabilities [5], on the basis of data coming from experts are Further, assuming that the system is in the reliability state subset {u,u+1,…,z} if all its subsystems are in this subset of reliability states, we conclude that the bulk cargo transportation system is a series system [4] of subsystems S 1 , S 2 , S 3 , S 6 , S 7 , S 8 and S 9 .
Under the assumption that changes of the bulk cargo transportation system operation states have an influence on both the subsystem i S reliability and the entire reliability structure [8], on the basis of expert opinions and statistical data given in [9], [10], the bulk cargo transportation system reliability structures and their components reliability functions at different operation states can be determined.
Additionally, we assume that subsystems , , are composed of four-state exponential components, with the reliability functions At the operation state 1 z , at loading of fertilizers from rail wagons on board the ship, the system is composed of seven non-homogenous series subsystems S 1 , S 2 , S 3 , S 6 , S 7 , S 8 , and S 9 forming a series structure.
The conditional reliability function of the system while it is at the operation state 1 z is given by (33)-(35) The expected values of the conditional lifetimes in the reliability state subsets at the operation state 1 z , calculated from the above result given by (33)-(35), are: and further, using (5), it follows that the conditional lifetimes in the particular reliability states at the operation state 1 z are: At the operation state 2 z , i.e. at the state of discharging rail wagons to storage tanks or hall, the system is built of three subsystems , 1 S 2 S and 3 S forming a series structure [4]. The conditional reliability function of the bulk cargo transportation system at the operation state 2 z is given by where (2) and further, using (5), it follows that the conditional lifetimes in the particular reliability states at the operation state 2 z are: At the operation state 3 z , i.e. at the loading of fertilizers from storage tanks or hall on board, the bulk cargo transportation system is built of six subsystems one seriesparallel subsystem S 4 and five series subsystems S 5 , S 6 , S 7 , S 8 , S 9 forming a series structure [4].
The conditional reliability function of the system while it is at the operation state 3 z is given by The expected values of the conditional lifetimes in the reliability state subsets at the operation state 3 z , calculated from the above result given by (41) and further, using (5), it follows that the conditional lifetimes in the particular reliability states at the operational state 3 z are: In the case when the system operation time is large enough, the unconditional reliability function of the bulk cargo transportation system is given by the vector where, according to (3) and after considering the values of , b p b = 1,2,3, given by (32), its co-ordinates are as follows:

.(48)
The mean values of the system unconditional lifetimes in the reliability state subsets, according to (4) are respectively: The mean values of the system lifetimes in the particular reliability states, (5), are If the critical reliability state is r = 2, then the system risk function, according to (6) (50) Hence, the moment when the system risk function (Fig. 1) exceeds a permitted level, for instance δ = 0.05, from (7), is τ = r -1 ( δ ) ≅ 0.000627 years.
(51) Figure 1 -The graph of the port bulk cargo transportation system risk function

OPTIMIZATION OF THE BULK CARGO TRANSPORTATION SYSTEM OPERATION PROCESS
In our case, as the critical state is 2 = r , then considering the expression for ) 2 ( μ , the objective function (8) , where i x and i x are lower and upper bounds of the unknown with the following bound constraints According to (15)-(17), we calculate and fix the optimal solution that maximizes linear function (55) according to the rule (19). Namely, we get . Finally, according to (21) after making the inverse to (53) substitution, we get the optimal transient probabilities 1 3 0.490

OPTIMAL RELIABILITY CHARACTERISTICS OF THE BULK CARGO TRANSPORTATION SYSTEM
Further, substituting the optimal solution (57) according to (24), we obtain the optimal solution for the mean value of the system unconditional lifetime in the reliability state subset }, 2 , 1 { } 3 { that respectively amounts: and according to (26), the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are Moreover, according to (24)-(25), the corresponding optimal unconditional multistate reliability function of the system is of the form ) , where according to (3) and after considering the values of b p , its co-ordinates are as follows: If the critical reliability state is r = 2, then the system risk function, according to (27) Hence, considering (28), the moment when the optimal system risk function (Fig. 2) exceeds a permitted level, for instance δ =0.05, is Comparing the bulk cargo transportation system reliability characteristics after its operation process optimization given by (58)-(62) with the corresponding characteristics before this optimization determined by (45)-(51) justifies this action.

OPTIMAL SOJOURN TIMES OF BULK CARGO TRANSPORTATION SYSTEM OPERATION PROCESS AT OPERATION STATES
Having the values of the optimal transient probabilities determined by (57), it is possible to find the optimal conditional and unconditional mean values of the sojourn times of the bulk cargo transportation operation process at the operation states and the optimal mean values of the total unconditional sojourn times of the bulk cargo transportation system operation process at the operation states during the fixed operation time as well.