1401

1 INTRODUCTION

This work focuses on the modeling and optimization of

operation costs for complex, multi-state, aging

technical systems, with particular emphasis on their

changing operation and safety states over time. The

main objective is to develop and apply cost models that

not only reflect the technical and economic realities of

system operation but also contribute to the improved

performance of real-world maritime and port

transportation systems.

A novel approach to cost modeling is presented,

which considers two distinct categories of system

states: operation states and safety states. Based on this

dual-state concept, two original cost models have been

proposed. The first, referred to as Model 1, represents

the total operation cost for fixed operation time,

incorporating both direct and indirect costs related to

maintenance, repairs, consumables, training, energy

usage, data protection, and system downtime. The

second, Model 2, quantifies the total cost incurred

while the system resides within the safety state subset,

focusing exclusively on the period from entry to exit

from the defined subset.

To support the safety-oriented classification of

system behaviour, five discrete safety states have been

introduced: full safety, high-level safety, medium-level

safety, low-level safety, and hazard state. The

transitions between these states depend on system-

specific characteristics, risks, and external factors, and

are managed by appropriate procedures or control

algorithms. The proposed cost models allow for the

assessment and optimization of expenditures under

varying safety conditions, enabling targeted analysis of

system behaviour within different safety regimes.

In industrial practice, optimizing the long-term

operation costs of such systems is crucial. Therefore,

linear programming techniques have been employed

Optimization of Operation Costs in Complex, Multi-

State, Aging Technical Systems

B. Magryta-Mut

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: This paper presents a methodological framework for the joint optimization of operation costs and

safety performance in complex, multi-state ageing technical systems. Two original models of system operation

cost are proposed: one based on total operation time, and the other on operation within safety state subsets. The

models are integrated with linear programming procedures and applied to a real-world maritime transport

system. The approach enables a balanced consideration of operational efficiency and system reliability, offering

a structured way to support decision making in the management of ageing infrastructure. The methodology relies

on both statistical operational data and expert knowledge. Potential extensions of this work include incorporating

external factors such as environmental conditions, as well as integrating repair, preventive maintenance, and

evolutionary multi objective optimization algorithms to enhance resilience and sustainability in critical technical

systems.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 19

Number 4

December 2025

DOI: 10.12716/1001.19.04.39

1402

to minimize the average total operation cost for fixed

time (Model 1) or while the system remains in safety

states not worse than a critical safety state (Model 2).

The optimization is based on the calculation of limit

transient probabilities - fundamental to Markov

processes - which approximate the long-run behaviour

of the system.

Furthermore, the study explores joint optimization

procedures that simultaneously address cost efficiency

and system safety. Two strategies are proposed: one

begins with cost minimization followed by safety

assessment, while the other prioritizes safety

maximization and then evaluates corresponding costs.

In both cases, system modifications are guided by

optimal limit transient probabilities, providing

operators with actionable strategies to balance

reliability, safety, and cost.

The proposed models and optimization procedures

offer a comprehensive framework for the economic

and safety analysis of complex systems, particularly in

critical infrastructure sectors. Their practical

application has the potential to enhance operation

efficiency and strategic decision-making in real-world

settings.

2 LITERATURE REVIEW

The literature on operation costs and optimization of

technical systems is largely grounded in the concept of

Life Cycle Costing (LCC) and optimization models in

maintenance management. The LCC approach, widely

introduced in the 1970s, involves analyzing costs

incurred throughout the entire life cycle of a technical

system. Early contributions to this field include [1], [2],

which define LCC as the total of direct, indirect,

recurring, and one-time costs associated with the

design, research, production, operation, and

maintenance of a system. The international standard

further develops this definition by detailing the life

cycle stages included in cost assessment frameworks

[3].

Three main approaches to LCC cost estimation are

commonly distinguished in the literature: parametric,

analogous, and detailed models, as outlined in [4], [5].

The inclusion of system reliability in LCC analysis is

emphasized in several studies [6]–[8], which highlight

the need to categorize costs into preventive (e.g.,

inspections) and failure related categories - both

internal and external. Such classifications are essential

in decision making regarding investment and system

operation.

In the domain of optimization models, the gap

between academic research and practical applications

is underscored in [9], [10], where the limited translation

of theoretical models into industrial use is discussed.

The need for more practice-oriented case studies is

further emphasized in [11]. Excessive focus on abstract

model development without regard to implementation

feasibility is criticized in [12]. Additionally, many

models optimize a single objective, which may not

reflect the multidimensional nature of real world

decision-making [13].

Research on multi criteria optimization - such as

that presented in [14]–[16] - suggests that practical

decision-making requires the simultaneous

consideration of multiple criteria, including cost,

reliability, and availability. A unifying optimization

framework integrating these criteria is discussed in

[17], highlighting the importance of not only

constructing models but understanding and aligning

them with organizational requirements.

In the context of maintenance strategies for multi

component systems, the complexity of component

interactions and the effects of imperfect maintenance

are addressed in [18]. The significance of component

interdependencies is examined in [19]–[21], while [22],

[23] develop preventive maintenance models that

incorporate continuous degradation and condition-

based indicators.

Finally, studies such as [24] focus on the economic

impact of imperfect repairs in critical infrastructure,

emphasizing the importance of optimal maintenance

planning to ensure reliability and minimize cost.

Existing research on the operation costs of complex,

multi-state, aging technical systems [25]–[28] has

primarily focused on the analysis and development of

optimization strategies aimed at minimizing operation

expenses while maintaining acceptable levels of

performance, reliability, and safety. This field is

particularly relevant in industries operating long-life

systems, where aging and degradation significantly

affect both maintenance and operational costs.

In summary, the literature review reveals that

research on life cycle costs and maintenance policy

optimization is essential for the effective management

of technical systems. The adoption of multi-criteria

approaches - taking into account cost, reliability,

availability, and business-specific requirements - is

indispensable for developing practical and efficient

management strategies for such systems.

3 OPERATION COSTS OF A GENERIC SYSTEM

In the literature, the concept of a generic system is

commonly used in systems theory to describe a

general, abstract model that can be adapted to a variety

of specific applications and domains. Such a system is

not directly tied to any particular implementation but

serves as a universal structure that can be modified

according to context and requirements. This concept

has been widely discussed in the scientific discourse,

including the work of Professor Krzysztof Kolbusz,

who emphasized its versatility in modeling various

processes and phenomena. Its generality facilitates the

application of the model across diverse operational

environments. In the present study, the term generic

system refers to the baseline subject of analysis - a

complex, multi state, aging technical system. The

chapter introduces a dedicated approach to modeling

the operational costs of such a system, taking into

account its inherent complexity and state-dependent

behaviour. Because the system under consideration

exhibits multiple degradation states over time, the

modeling of operational costs must also incorporate

transitions between different safety states. These safety

states represent the system’s varying levels of

operational security and reliability throughout its life

cycle. While the primary focus of this study is the

optimization of operational costs, integrating safety-

1403

related aspects is essential to achieve a more

comprehensive and realistic assessment. By jointly

considering costs and safety dynamics, the model

provides a more robust foundation for decision-

making in the management of complex technical

systems. The analysis considers two types of states:

operation states and safety states. It also allows for the

evaluation of different conditional costs, depending on

whether a specific operational or safety state is fixed.

Based on this framework, two original models of

system operational costs have been proposed: Model 1

– System operation cost model for fixed operation time,

Model 2 – System operation cost model in safety state

subsets.

3.1 System operation cost model for fixed operation time

Model 1 is a mathematical representation of all costs

associated with the maintenance and operation of a

given system over a specified period. This model

accounts for both direct and indirect costs incurred

during system operation. Direct costs refer to expenses

that are directly related to system maintenance and

usage within the considered time interval, such as

service and repair costs, the cost of consumable

materials, or personnel training costs. In contrast,

indirect costs are those not directly tied to system

operation but result from its usage. These may include

electricity consumption, expenses related to data

protection and cybersecurity, and potential costs

associated with operational delays.

The total operational cost of the system over the

time interval 𝜃

( ) ( )

( )



  



=





(1)

where pb, b = 1,2,...,ν, are limit transient probabilities at

operation states, and

( )

, 0,

t dt





=





b = 1,2,...,ν, (2)

are the total values of the system total conditional

operation costs at the particular system operation

states during the operation time θ, [27]–[28].

3.2 System operation cost model in safety state subsets

Model 2 is a mathematical representation of all costs

associated with the maintenance and operation of the

system while it resides within a defined subset of safety

states from the moment the system enters the subset

until it exits. This model includes both direct and

indirect costs incurred specifically during the system’s

presence in the safety state subsets. Costs associated

with other states that do not belong to the defined

safety subset are not included in the analysis. The

definitions of direct and indirect costs used in this

model are analogous to those in Model 1.

The expected total operational cost of the system

within safety states is defined by a vector:

( ) ( ) ( ) ( )

,,2 ,.

..1 ,

ˆˆ



=



C C C C

, (3)

with components that are expected values of the total

system operating costs in safety states given by linear

equations

( ) ( )

( )

]

[

u p u

  



, u = 1,2,...,z, b = 1,2,…,ν. (4)

Where, analogously to the previous model, pb, b =

1,2,…,ν, are limit transient probabilities at operation

states.

The values

( )

u t u dt









=





u = 1,2,...,z,

b = 1,2,...,ν. (5)

are the mean total operation costs of the system in

safety states while it resides in individual operation

states [27]–[30].

4 OPTIMIZATION APPROACH

For the purpose of cost optimization, linear

programming has been proposed to minimize the total

operational cost for fixed operation time, as well as to

minimize the total operational cost in the safety state

subsets not worse than the critical safety state based

respectively on Cost Model 1 and Cost Model 2. Similar

to the approach applied in safety optimization, the

optimization procedures here make use of previously

established results related to safety performance. This

integration enables a consistent and unified framework

for optimizing both economic and safety aspects of

system operation. [28], [31]

The optimization of operation costs in a multi-state,

aging technical system can be effectively addressed

through linear programming, where the key decision

variables are the limit values of transient probabilities

of the system residing in particular operational states.

[28], [31].

4.1 Step 1: Definition of the Cost Function

The total unconditional operation cost of the system

over a fixed operation time is modeled as a weighted

sum of conditional operation costs associated with

each operational state. The weights are the values of

limit transient probabilities pb, b = 1,2,...,ν, of the system

operation process at the operation states, and the costs

( )

, 0,









are considered to be fixed and known.

4.2 Step 2: Formulation of the Linear Programming

Problem

The goal is to minimize the total unconditional

operation costs by selecting an optimal probability

distribution under the following constraints:

− The probabilities must sum to 1 (normalization

condition),

− Each probability pb must remain within a defined

lower and upper bound

 

− All costs

( )







must be non-negative.

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This problem leads to a standard linear

programming formulation where the objective

function is linear in the probabilities.

4.3 Step 3: Variable Transformation for Computational

Efficiency

To streamline the computation, the conditional cost

values are first sorted in non-decreasing order, and the

corresponding probabilities are relabeled accordingly.

That is, new variables xi are introduced to represent the

reordered

, along with corresponding bounds

and

4.4 Step 4: Allocation Strategy

To solve the problem, an efficient allocation strategy is

used based on the structure of the linear objective

function:

1. Calculate the remaining probability mass

x x y x

= = −



and

0, 0xx==

and

x x x x



for I = 1,2,...,ν.

2. Identify the largest index III for which the

accumulated width

x x y−

3. Depending on the value of III, the optimal solution

is constructed as:

− Case if I = 0, the optimal solution is

x y x=+

and

xx=

for i = 2,3,…,ν,

− Case if 0 < I < v, the optimal solution is

xx=

for

i=1,2,...,I,

x y x x x

= − + +

and

xx=

for

i=I+2,I+3,...,ν;

− Case I = v: assign all probabilities to their upper

bounds.

4.5 Step 5: Back-Transformation and Interpretation

After determining the optimal values xi, they are

transformed back to the original probability variables

pb according to the earlier index mapping. This yields

the optimal steady-state distribution that minimizes

the expected total operation cost.

A similar procedure is applied when optimizing the

operation cost within selected safety state subsets. In

this case, the objective is to minimize the expected cost

incurred only while the system operates within a

subset of acceptable safety states, typically starting

from a critical safety threshold r. The same steps,

formulating the linear cost function, defining bounds,

sorting, transforming variables, allocating probability

mass, and performing inverse substitution are

repeated, but now with the cost function and

constraints restricted to the safety state subsets. [28],

[30]-[31]

5 JOINT OPTIMIZATION OF OPERATION COST

AND SAFETY IN MULTI-STATE AGING

TECHNICAL SYSTEMS

The joint optimization of system operation cost and

safety in complex, multi-state, aging technical systems

involves an integrated approach that leverages the

relationship between the system's probabilistic

behaviour and its economic and safety performance

metrics [28], [32]-[35]. This approach is centered on the

determination of optimal limit transient probabilities

of the system’s operation process in its various states,

which serve as the decision variables linking both cost

and safety analyses. When the goal is to determine the

operation cost corresponding to the system's

maximum safety, the first step is to apply a general

safety model along with a dedicated safety

optimization procedure. This enables the identification

of optimal limit transient probabilities that maximize

the system’s safety indicators such as extended

residence time in safe states, minimal exposure to risk,

or highest resilience. Once these optimal probabilities

are obtained, they are substituted into the previously

developed cost models. This substitution yields the

expected total operational cost over a fixed operation

period or within a predefined subset of safety states not

worse than a critical level. The result is a quantification

of the system's operational costs under the assumption

of its best achievable safety configuration. This

methodology allows for a consistent transition from

safety optimization to cost evaluation, using a shared

probabilistic foundation. Conversely, when the

objective is to evaluate the safety characteristics

corresponding to the system's minimal operation cost,

the optimization process begins with the application of

the operation cost model. Through linear

programming techniques, the limit transient

probabilities that minimize the system’s total cost

either over a specified time interval or within a critical

safety region are determined. These cost optimal

probabilities are then inserted into the safety function

formulations, enabling the calculation of the

conditional safety indicators associated with the lowest

cost configuration. This process makes it possible to

assess how minimizing costs affects the system’s risk

exposure, operational security, and long term

reliability. In both approaches, the key lies in the use of

limit transient probabilities as a common decision

vector. Whether derived through safety or cost

optimization, these probabilities can be propagated

through the complementary domain's model to obtain

the full profile of system behaviour. This bidirectional

strategy creates a unified decision support framework

that quantifies the trade-offs between safety and cost.

It enables operators to answer critical questions, such

as how much additional cost must be incurred to

achieve a desired safety level, or what safety risks are

introduced by stringent cost minimization. Ultimately,

the joint optimization procedure contributes to the

effective and rational management of aging, multi-

state systems, particularly in safety-critical sectors. It

offers a foundation for designing operational strategies

that simultaneously satisfy economic constraints and

maintain acceptable levels of safety, making it a

valuable tool in infrastructure management, transport

operations, and industrial system engineering.

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6 CASE STUDY: OPERATIONAL COST AND

SAFETY MODELING OF A MULTI-STATE

FERRY SYSTEM

6.1 System description

In this case study, we examine the operation and cost

structure of a passenger ferry operating daily on the

Gdynia–Karlskrona route across the Baltic Sea. The

ferry constitutes a complex technical system composed

of several interdependent subsystems that significantly

influence its operational safety and performance.

Figure 1. Ferry connection between Gdynia and Karlskrona

The ferry’s technical infrastructure is modeled as a

system of interconnected subsystems: navigation (S1),

propulsion and steering (S2), loading and unloading

(S3), stability control (S4), anchoring and mooring (S5),

as well as protection and rescue (S6) and social services

(S7). For the purposes of this analysis, only the core

technical subsystems S1– S5 are considered, collectively

referred to as the technical system of the ferry. Each

subsystem is hierarchically decomposed into lower

level components (e.g., main engines, steering

thrusters, gangways), and the overall safety structure

is modeled as a series system in which the failure of any

active subsystem implies a degradation of the system’s

operational safety. The ferry’s operation is represented

by a stochastic process consisting of 18 defined

operational states which reflect the complete voyage

cycle, from docking, loading, and departure in Gdynia,

through navigation in various maritime zones, to

arrival and unloading in Karlskrona, and vice versa.

The process follows a sequential and cyclic structure

and is characterized by limit transient probabilities,

empirically derived from real voyage data owing to the

high frequency and regularity of the ferry’s operation

[28], [30], [32], [34]-[35].

To assess the operational behavior of the ferry

system, two primary sets of characteristics are used:

The limit transient probabilities of being in each

operational state (e.g., p5=0.363, p13=0.351), which

indicate that the ferry spends the majority of its time

navigating in open waters under Polish and Swedish

jurisdiction.

The approximate mean values of total sojourn times

of the ferry in each state during a fixed time period (θ

= 30 days = 720 hours), derived from Mb=pb

⋅

θ, which

allows for time based allocation of operational costs.

In the cost analysis, each subsystem is assigned a set

of state-dependent hourly cost rates, differentiated by

active usage versus standby conditions. For instance,

the navigation subsystem S1 incurs a constant

operational cost of 20c when active and 10 c otherwise.

Similarly, the propulsion system S2 has differentiated

costs depending on whether it operates in restricted or

open waters (75c or 55c, respectively), while its standby

cost is set at 25c. c is a coefficient, adopted arbitrarily,

used to determine the costs of subsystems. Other

subsystems follow similar differentiated costing

schemes.

6.2 Results for Model 1

A comparison of the ferry's operation cost results based

on Model 1 representing the total operation cost for

fixed operation time is presented in Table 1., [28], [32]-

[35].

Table 1. Comparison of the ferry system’s total operation

cost before and after optimization based on Model 1

Model 1 – cost

for fixed

operation time

Before

optimization

After

optimization

Joint

optimization

19 490.69c

17 056.54c

19 308.006c

100%

87.5%

99%

The optimization procedure led to a reduction in

total operational costs of approximately 12.5%.

Furthermore, the conditional total operation cost of the

ferry, calculated under the assumption of maximum

system safety, is found to be lower than the before

optimization cost associated with the unoptimized

safety profile, yet higher than the minimum cost

obtained through direct cost optimization. This

outcome implies that if the primary objective is to

maintain a high level of operational safety rather than

to minimize operational expenses, the system's

operation process can be modified accordingly. In

particular, the limit transient probabilities within the

cost model can be replaced by their optimal values

derived from the safety optimization procedure. This

allows for the adjustment of the expected residence

times of the system in particular states, supporting

safety-oriented operational strategies while keeping

costs within acceptable bounds.

A comparison of the ferry’s safety performance

results is presented in Table 2., [28], [32]-[35].

Table 2. Safety indicator values

Model 1 –

safety

indicator

values

Before

optimization

After

optimization

Joint

optimization

The expected

values in the

safety state

subsets

{1, 2, 3, 4}

1.694

1.697

1.6923

{2, 3, 4}

1.395

1.397

1.39364

{3, 4}

1.244

1.247

1.243

{ 4}

1.114

1.115

1.11318

The moment,

when system

risk function

exceeds a

permitted

level

0.073

0.04634

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The mean

intensities of

aging

{1, 2, 3, 4}

0.590363

0.589275

0.59091

{2, 3, 4}

0.716869

0.715819

0.71755

{3, 4}

0.803573

0.801925

0.8045

{ 4}

0.897470

0.869565

0.89833

The

coefficients

of the

operation

process

impact

intensities of

ageing

{1, 2, 3, 4}

1.044942

1.043

1.04591

{2, 3, 4}

1.058098

1.057

1.0591

{3, 4}

1.044645

1.043

1.04585

{ 4}

1.044655

1.043

1.04566

The

coefficient of

resilience to

the operation

process

impact

{1, 2, 3, 4}

95.70 %

95.88 %

95.61 %

{2, 3, 4}

94.51 %

94.61 %

94.42 %

{3, 4}

95.73 %

95.88 %

95.62 %

{ 4}

95.73 %

95.88 %

95.63 %

The expected values (expressed in years) in the

safety state subsets ({1,2,3,4}, {2,3,4}, {3,4}, {4}) show

minor increases after safety optimization, indicating a

slightly enhanced presence of the system in safer

operational states. However, in the joint optimization

scenario, these expected values are marginally reduced

compared to post-safety optimization results. For

example, in the critical safety state {4}, the expected

value decreases from 1.115 (after safety optimization)

to 1.11318 in joint optimization. This suggests that

while joint optimization introduces minimal trade-offs

in terms of safety positioning, the degradation is

negligible and may be acceptable when cost reduction

is also a priority. A significant improvement is

observed in the moment when the system risk function

exceeds a permitted level. While this threshold remains

constant at 0.073 year before and after safety

optimization, joint optimization leads to a noticeable

delay in risk emergence, reducing the threshold to

0.04634. This outcome indicates that the combined

optimization of cost and safety results in a more robust

operational profile, potentially deferring critical risk

exposure.

The mean intensities of aging generally decline after

safety optimization, reflecting a reduction in the rate of

system degradation. For instance, in safety state {4}, the

aging intensity decreases from 0.897470 to 0.869565.

However, under joint optimization, these intensities

slightly increase again (e.g., to 0.89833 for {4}),

reflecting the partial reallocation of optimization focus

toward cost efficiency. This pattern illustrates the

delicate balance between maintaining system

longevity and achieving economic objectives. The

coefficients of the operation process impact on aging

intensities mirror this behavior. Post safety

optimization results show a marginal reduction in

values across all state subsets, while joint optimization

slightly increases them, exceeding even the pre

optimization baseline in some cases (e.g., {2,3,4}: from

1.058098 to 1.0591). Although the increases are modest,

they suggest that the cost efficiency gains of joint

optimization are accompanied by slightly intensified

aging processes. Despite this, the coefficient of

resilience to the operation process impact, a key safety

metric, remains relatively stable and high across all

conditions. While resilience improves after safety

optimization (e.g., from 95.70% to 95.88% in {1,2,3,4}),

it only marginally decreases under joint optimization

(to 95.61%). This indicates that even with increased

operational efficiency, the system retains a strong

capacity to withstand the effects of aging under the

influence of operational conditions.

6.3 Results for Model 2

A comparison of the ferry's operation cost results based

on Model 2 representing the total operation cost in the

safety state subsets is presented in Table 3., [28], [32]-

[35].

Table 3. Comparison of the ferry system’s total operation

cost before and after optimization based on Model 2

Model 2 –

cost in safety

state subsets

Before

optimization

After

optimization

Joint

optimization

{1, 2, 3, 4}

175.15054c

173.034c

{2, 3, 4}

144.13643c

142.433c

{3, 4}

128.71551c

127.060c

{ 4}

115.24016c

113.795c

In each of the analyzed safety state subsets ({1,2,3,4},

{2,3,4}, {3,4}, and {4}), a reduction in the total operation

cost is observed after the optimization process. This

confirms the effectiveness of the proposed

optimization approach in reducing system operation

costs while maintaining a predefined safety level. The

operation costs after joint optimization, i.e., the

integrated approach to simultaneously optimizing

safety and cost, are equal to the costs obtained through

safety-focused optimization alone. This suggests that

in this case, the safety constraints were dominant in the

optimization process, meaning the minimization of

costs did not compromise the safety levels within the

considered state subsets. A compromise was achieved

without deterioration in safety performance. The

largest absolute cost reduction in the safety state subset

({1,2,3,4}), where the cost decreased from 175.15054c to

173.034c. Although the relative change is modest

(approximately 1.2%), it demonstrates that even within

restricted operational modifications, meaningful cost

savings can be realized.

A comparison of the ferry’s safety performance

results is presented in Table 4., [28], [32]-[35].

Table 4. Safety indicator values

Model 2 –

safety

indicator

values

Before

optimization

After

optimization

Joint

optimization

The expected

values in the

safety state

subsets

{1, 2, 3, 4}

1.694

1.697

1.69629

{2, 3, 4}

1.395

1.397

1.39654

{3, 4}

1.244

1.247

1.24527

{ 4}

1.114

1.115

1.11535

The moment,

when system

risk function

exceeds a

permitted

level

0.073

0.06467

The mean

intensities of

aging

{1, 2, 3, 4}

0.590363

0.589275

0.58952

{2, 3, 4}

0.716869

0.715819

0.71605

{3, 4}

0.803573

0.801925

0.80304

{ 4}

0.897470

0.869565

0.89658

The

coefficients

of the

operation

process

impact

intensities of

ageing

{1, 2, 3, 4}

1.044942

1.043

1.04345

{2, 3, 4}

1.058098

1.057

1.05689

{3, 4}

1.044645

1.043

1.04358

{ 4}

1.044655

1.043

1.04362

1407

The

coefficient of

resilience to

the operation

process

impact

{1, 2, 3, 4}

95.70 %

95.88 %

95.83 %

{2, 3, 4}

94.51 %

94.61 %

{3, 4}

95.73 %

95.88 %

95.82 %

{ 4}

95.73 %

95.88 %

95.82 %

The expected values in safety state subsets show

marginal improvements after both safety targeted and

joint optimization. For each considered subset

({1,2,3,4}, {2,3,4}, {3,4}, and {4}), a slight increase in the

expected value is observed, indicating a higher

likelihood of the system remaining in safer states for

longer periods. The joint optimization results are

nearly identical to those achieved after safety specific

optimization, suggesting a negligible compromise in

safety due to the inclusion of cost factors. The moment

when the system risk function exceeds the permitted

level remains constant at 0.073 before and after safety

only optimization but decreases to 0.06467 under joint

optimization. This indicates that joint optimization not

only preserves safety performance but slightly

improves system resilience by delaying the onset of

critical risk thresholds. The mean intensities of aging

decrease slightly after safety optimization and are

maintained at nearly the same levels under joint

optimization. For example, in the critical state subset

{4}, the intensity drops from 0.897470 before

optimization to 0.869565, and then marginally

increases to 0.89658 in joint optimization. These values

reflect reduced degradation rates of the system

components during optimized operation, which

implies improved long term reliability. The coefficients

of the operation process impact on intensities of aging

are consistently reduced after optimization across all

state subsets, again with negligible differences between

safety only and joint optimization results. This

suggests that the optimization process effectively

minimizes the operational load contributing to system

wear. The coefficients of resilience to the operation

process impact improve slightly after safety

optimization and are preserved under joint

optimization. The values range from 94.51% to 95.88%,

indicating a high level of system robustness. For

instance, in the {1,2,3,4} subset, resilience increases

from 95.70% to 95.88%, and is maintained at 95.83% in

joint optimization.

7 CONCLUSIONS

The article presents original models and optimization

procedures for assessing and improving both the

operation costs and safety levels of complex, multi-

state, ageing technical systems. These models were

applied to the technical system of a Baltic Sea ferry

operating between Gdynia and Karlskrona. Cost and

safety evaluations were based on detailed operational

data and expert derived estimations of parameters

related to safety and subsystem operation costs. The

only significant limitation in practical application of

the models is the availability of sufficiently accurate

statistical data from system users. The study shows

that optimized ferry operation reduces costs by over

12%, while maintaining satisfactory safety levels.

Future research could explore the influence of non-

operational factors, such as climatic or weather

conditions [36], on system reliability and cost.

Additionally, expanding the optimization framework

to include repair strategies [37], preventive actions [22],

and system corrections could enable more holistic

infrastructure management. The use of multi criteria

optimization methods and evolutionary algorithms is

also recommended [38], especially for managing

ageing systems with irreversible physical and chemical

degradation [39].

ACKNOWLEDGMENT

This work was developed within the framework of the grant

no. WN/PI/2025/05 sponsored by the Gdynia Maritime

University, Poland.

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