Global Spare Parts Exploitation Costs Optymization and Its Reduction

to Rectangular Knapsack Problem

Andrzej Chmielowiec

1 a

, Leszek Klich

1 b

, Weronika Wo

1 c

and Adam Błachowicz

2 d

Faculty of Mechanics and Technology, Rzeszow University of Technology, Kwiatkowskiego 4, Stalowa Wola, Poland

Pilkington Automotive Poland, Strefowa 17, Chmiel

ow, Poland

Keywords:

Maintenance, Cost Optimization, Weibull Distribution, Exploitation Costs, Manufacturing Execution System,

Knapsack Problem.

Abstract:

Maintenance costs represent a signiﬁcant portion of the budget in large enterprises. Unplanned downtime and

breakdowns have a very negative impact on the ﬁnancial result. Minimizing their number is today a huge chal-

lenge for systems supporting production management and maintenance processes. Due to intensive research

in the ﬁeld of reliability today, we know a lot about the optimal use of a single spare part. Unfortunately, the

analytical methods used for this purpose do not work well in the case of simultaneous analysis of the entire

production line. The article proposes a combination of analytical methods and discrete optimization methods

for global management of spare parts operating costs. The purpose of the presented algorithm is to support

decision-making that minimize the global cost of maintaining reliability in the entire production company.

1 INTRODUCTION

Despite the installation of an increasing number of

sensors and monitoring the degree of wear of many

systems and their subassemblies, there is still a large

number of elements which usability status is de-

termined in a binary manner: functional / non-

functional. An example is an oil hose for which the

performance level on the ﬂy cannot be determined.

Perhaps in the future, advanced vision systems will be

able to automatically detect microcracks on its surface

and track wear levels. At present, however, the opera-

tion of such cases is purely reactive. The exchange

is made at a strictly deﬁned moment or the failure

is removed when it occurs. From the point of view

of maintenance people, this is a rather uncomfortable

situation. However, for each such element on the pro-

duction line, we can determine the optimum working

time. Optimum time is understood here as the time

that minimizes the costs of servicing. Intuition sug-

gests that if the removal of a failure is much more

expensive than a scheduled replacement, the standard

working time of such an element should be shorter

https://orcid.org/0000-0001-6629-0029

https://orcid.org/0000-0001-6099-2417

https://orcid.org/0000-0003-4709-4369

https://orcid.org/0000-0003-4465-2652

than in a situation where the costs of removing a fail-

ure and scheduled replacement are similar. However,

the question is: when should the replacement be made

so that the long-term costs are as low as possible?

This problem will be discussed in this article. In the

following parts, it will be shown how modern IT tools

combined with the classical theory of reliability can

support the decision-making process in a manufactur-

ing company.

The Figure 1 shows the Weibull distribution with

parameters of α = β = 5, respectively. This distri-

bution was chosen as an example to illustrate the re-

liability of aging components. In practice, for each

element, the α and β parameters should be estimated

based on historical data (which will be presented in

the following sections) or information from the man-

ufacturer. It can be calculated that the average prod-

uct lifetime for a distribution with these parameters is

about 4.6 time units (e.g. months).

The Figure 1 shows that the systematic replace-

ment of such an element every one time unit prac-

tically eliminates the problem of failure. Therefore,

the manufacturer may recommend that users regularly

replace them at a speciﬁc time period. However, is

this approach economically justiﬁed? Unfortunately,

there is no clear answer to this problem. The solu-

tion depends on the cost of replacing this element dur-

ing standard service activities, and the cost of replace-

364

Chmielowiec, A., Klich, L., Wo

s, W. and Błachowicz, A.

Global Spare Parts Exploitation Costs Optymization and Its Reduction to Rectangular Knapsack Problem.

DOI: 10.5220/0011067400003179

In Proceedings of the 24th International Conference on Enterprise Information Systems (ICEIS 2022) - Volume 1, pages 364-373

ISBN: 978-989-758-569-2; ISSN: 2184-4992

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10

Failure PDF

Time

α=5, β=5

Figure 1: Probability density for the Weibull distribution

with α = 5 and β = 5.

ment in the event of a failure. It should be emphasized

that the costs related to downtime should be added to

the cost of the service replacement as well as to the

cost of the replacement during a failure. This is a very

important cost component that can cause the same el-

ement used at different points in the production line

to be treated differently. In fact, any ﬁnite resource at

the company’s disposal that has any value can be con-

sidered as a cost. This can be time, money, energy,

etc. In principle, cost should be understood to mean

all the necessary resources that are consumed during

replacement (both service replacement and failure re-

placement).

The problem of age replacement policy is the sub-

ject of many authors’ publications. First (Barlow and

Proschan, 1965) and (Barlow and Proschan, 1975)

have analyzed the optimization of the expected op-

erating costs and the impact of the replacement pol-

icy on these costs. Moreover, for several common

failure distributions methods have been developed by

(Glasser, 1967). On the other hand, (Fox, 1966) and

(Ran and Rosenlund, 1976) consider the optimal re-

placement time, taking into account discounting - this

issue is especially important in the case of rapidly

changing prices. The publications (Scmeaffer, 1971),

(Cl

eroux and Hanscom, 1974), (Cl

eroux et al., 1979)

and (Subramanian and Wolff, 1976) present a gener-

alized approach to the optimization of operating costs

and preventive replacement. The conﬁdence intervals

with unknown failure distributions for the problem of

minimizing the expected replacement costs were also

quite intensively investigated, which can be found in

the articles (Ingram and Scheaffer, 1976), (Frees and

Ruppert, 1985), (L

eger and Cl

eroux, 1992). Next,

(Berg and Epstein, 1978) compares an age-related ex-

change policy with exchange policy at strictly deﬁned

periods. The articles (Christer, 1978) and (Ansell

et al., 1984) analyze the asymptotic operating costs

assuming the replacement policy according to the age

of the element. A very extensive bibliography on the

problem of optimization of maintenance costs was

presented by (Nakagawa, 2003). In the review arti-

cle (Zhao et al., 2017) an up-to-date summary of the

state of knowledge in this ﬁeld and perspectives for

further research directions can be found. It should

be emphasized that the problem of the replacement

policy as a result of aging is also valid today, as evi-

denced by publications from recent years (Knopik and

Migawa, 2017), (Fu et al., 2018), (Finkelstein et al.,

2020), (Wang et al., 2021a), (Zhao et al., 2021). One

problem in the practical use of the methods devel-

oped so far is focusing attention on a single element

or spare part. In this article, a global approach will

be proposed, the aim of which will be to optimize the

expected operating costs of the entire production line.

For this reason, under certain assumptions, analyti-

cal methods will be combined with discrete optimiza-

tion methods reducing the problem to a two- or one-

dimensional knapsack problem.

2 MINIMAL OPERATING COST

FOR SINGLE ELEMENT

The problem of optimal selection of the replacement

time will be presented in a more formal way. Let C

be the cost of removing the failure of a given element,

and C

the cost of the scheduled service replacement.

Moreover, let a = C

. In addition, it is assumed

that C

= C

+ τ C

, where C

is the downtime cost

per unit of time and τ is the downtime required dur-

ing a failure. It follows from the assumptions that the

coefﬁcient a > 1. It is then assumed that there will

be regular maintenance replacements for this item at

exactly T units of time since the previous replace-

ment. If F(t) is the failure probability at time t, and

G(t) = 1−F(t) is the reliability at time t, then the ex-

pected cost of service activities for the given replace-

ment time T is estimated as

= C

F(T ) +C

G(T ). (1)

In the formula above, the cost C

is simply the

weighted average cost of failure C

that occurs with

probability F(T ) and service cost C

that occurs with

probability G(T ) = 1 − F(T ). If we consider that

= a ·C

, we get the following equality

= C



1 + (a − 1)F(T )



. (2)

The calculated cost C

is not enough. It does not

yet take into account the frequency with which such

changes will be made. There may be a situation in

which the average cost of C

is small, but the fre-

quency with which it will be borne is so high that it

is not an optimal solution. So let µ

denote the mean

time between replacements of a given element, both

Global Spare Parts Exploitation Costs Optymization and Its Reduction to Rectangular Knapsack Problem

365

due to the failure and the planned service replace-

ment. Then the value µ

−1

determines the frequency

of replacement of the element and allows to deﬁne

the ε

function which expresses the relative cost of

maintaining the efﬁciency of the element per unit of

time

· µ

−1

. (3)

Minimizing the ε

function is our main task. Before

we get down to that, it is necessary to deﬁne the for-

mula for the expected replacement time µ

. It is given

by the following formula (Nakagawa, 2006)

t dF(t) + T G(T ) =

G(t) dt. (4)

Thus, the ε

function deﬁning the relative cost of

maintaining efﬁciency per unit of time is expressed

as follows

1 + (a − 1)F(T )

G(t) dt

. (5)

Nakagawa (Nakagawa, 2003) showed that T minimiz-

ing the value of ε

meets one of the following condi-

tions:

1. is ﬁnite and is given uniquely,

2. is inﬁnite.

0 1 2 3 4 5 6 7 8 9 10

Relative cost of maintaining efficiency

per unit of time ε

Time

a=2

a=4

a=8

a=16

a=32

Figure 2: The functions of the relative cost of maintaining

efﬁciency in the time unit ε

for example values of the a

parameter and the Weibull distribution with the parameters

α = β = 5.

It should be noted that in the absence of a ﬁnite value

of T , all service replacements are unproﬁtable. In

such cases, the only economically sensible strategy

is to wait for a failure.

The introduced concepts will be illustrated on an

exemplary Weibull distribution. The Figure 2 shows

the graphs of the ε

function for a few example values

of a and the Weibull failure probability

F(t) = 1 − e

−(t/α)

, (6)

with the following parameters: α = β = 5. It can be

Table 1: Replacement times that minimize the relative cost

of maintaining the device in service for the speciﬁed values

of a and the Weibull distribution of α = β = 5.

a α β Replacement time

to minimize maintenance costs

2 5 5 3.80

4 5 5 3.05

8 5 5 2.57

16 5 5 2.21

32 5 5 1.91

seen from these graphs that the function of the relative

cost of maintaining efﬁciency takes its minimum the

sooner the higher the value of the a coefﬁcient. This

is in line with the intuition that the higher the cost of

failure compared to the cost of service replacement,

the shorter the time should be the component used. If

the cost of removing the failure is twice as much as the

cost of servicing, the element should be replaced after

3.8 units of time. When the cost of removing a failure

exceeds four times the cost of regular service, the ele-

ment should be replaced after 3.05 time units. A sum-

mary of all values for the above-mentioned graph is

given in the Table 1.

0 2 4 6 8 10 12 14 16

Service replacement period T

minimizing the cost ε

Breakdown cost to service cost ratio a = c

/ c

min(ε

)

Figure 3: Values of T minimizing the relative cost of main-

taining efﬁciency in the time unit ε

for the given values of

the parameter a.

Figure 3 shows how the maximum lifetime of

a given component changes if the relative cost of

failure increases. The presented ones concerned the

Weibull distribution with the parameters α = β = 5. If

the aging process of a given element is approximated

by the Weibull distribution with different values of

these parameters, then the entire analysis should be

performed again for the modiﬁed data.

ICEIS 2022 - 24th International Conference on Enterprise Information Systems

366

3 EXAMPLES OF

OPTIMISATION FOR REAL

INDUSTRIAL ELEMENTS

For the purposes of this article, certain parts have been

selected. They are: membrane, swivel joint, bando

belt and IR belt. This parts are shown in Figure 4.

a) Membrane b) Swivel joint

c) Bando belt d) IR belt

Figure 4: Industrial spare parts used as examples for main-

tenance cost optimisation.

The data on the failure-free operation time of individ-

ual spare parts can be found in the Table 2.

Table 2: Uptime in days for each industrial spare part.

Spare part Uptime (days)

Membrane 1, 40, 45, 16, 6, 29, 6, 76, 14,

34, 58, 91, 49, 42, 156, 28, 35,

88, 96, 106, 13, 17

Swivel joint 72, 208, 51, 153, 245, 119, 41,

692, 79, 235, 99, 342

Bando belt 14, 55, 107, 44, 146, 63, 18,

78, 42, 41, 40, 28, 59, 7, 106,

48, 24, 66, 23, 50, 3, 67, 7, 43,

72, 1, 75, 37, 27, 49, 34, 49, 6,

74, 91

IR belt 130, 373, 172, 178, 272, 138,

303, 85

They were used to determine the parameters for the

Weibull distribution using the variable linearization

method and the least squares method (Lai et al.,

2006), (Lai, 2014). The approximation results were

illustrated with probability graphs for each of the an-

alyzed spare parts.

# Import libraries.

from reliability.Distributions import \

Weibull_Distribution

from reliability.Fitters import Fit_Weibull_2P

import matplotlib.pyplot as plt

# Set uptime data.

data = [85,130,138,172,178,272,303,373]

# Prepare Probability Plot using least

# square method. It also determine Weibull

# distribution parameters.

fit = Fit_Weibull_2P(failures=data, \

method=’LS’, linewidth=2.5)

# To get details about determined distribution

# use properties and functions of

# fit.distribution

# object. It is possible to get both plots and

# numeric data.

# ... Add plot options and show ...

plt.show()

In this part, the use of open Python libraries for mod-

eling the life expectancy of exemplary spare parts

used in industry will be presented. For more in-

formation on using Python libraries to analyze and

parameterize the Weibull distribution, see the article

(Chmielowiec and Klich, 2021). The obtained results

will be the basis for determining the optimal operat-

ing times for each part. They can be considered as

the ﬁrst level to support the parts replacement man-

agement process.

It should be noted that the choice of the Weibull

distribution as a distribution modeling the life cy-

cle of spare parts used in industry is not accidental.

Since the introduction (Weibull, 1939), this distri-

bution is widely used in numerous industrial issues:

strength of glass (Keshevan et al., 1980), pitting cor-

rosion (Sheikh et al., 1990), adhesive wear of met-

als (Queeshi and Sheikh, 1997), failure rate of shells

(Almeida, 1999), failure frequency of brittle materi-

als (Fok et al., 2001), failure rate of composite ma-

terials (Newell et al., 2002), wear of concrete ele-

ments (Li et al., 2003), fatigue life of aluminum alloys

(Hemphill et al., 2012), fatigue life of Al-Si castings

(Aigner et al., 2018), strength of polyethylene tereph-

thalate ﬁbers (Xie et al., 2020), failure of joints under

shear (Bai et al., 2020) or viscosity-temperature rela-

tion for damping ﬂuids (Chmielowiec et al., 2021).

On the basis of ﬁnancial data, the optimal times

for replacing individual elements were also deter-

mined. Although these times optimize the operating

costs of each part separately, they do not allow to look

at the problem from a global point of view – there

are hundreds of such elements functioning in a man-

ufacturing company. Therefore, in the next the use of

rectangular knapsack problem algorithms to support

the management process of maintenance costs will be

discussed.

Global Spare Parts Exploitation Costs Optymization and Its Reduction to Rectangular Knapsack Problem

367

3.1 Membrane

The membrane is one of the key components of the

pneumatic valve actuator, which is used to create

a vacuum in an electric press. The press and the valve

mounted in it operate in cycles of approximately 20

seconds. During this time, the membrane is subjected

to forces causing its systematic wear. A rupture in

the membrane causes the production line to stop com-

pletely for τ

= 4 h. The cost of replacing an element

is C

M,1

= 200 EUR, and the cost of stopping a line is

approximately C

D,1

= 650 EUR h. Therefore, the cost

of an emergency replacement of this element should

be estimated at C

F,1

= C

M,1

+ τ

D,1

= 2 800 EUR.

This means the a ratio for the membrane is 14.0.

Figure 5: Membrane Weibull probability plot determined

by coordinate linearization and less square method.

Using the data on the failure rate of the element,

it is possible to estimate the parameters of the dis-

tribution. The Figure 5 shows the method of deter-

mining the parameters using the least squares method.

The parameters obtained by this way have the value

= 51.559 and β

= 1.021. The Weibull distribu-

tion density plot for such parameters is presented in

Figure 6. Determining the optimal replacement time

as a value that minimizes the ε

function given by

the equation (5), we get T

= 980 days. Observing

the historical data, it can be concluded that in practice

it means replacement only in the event of a failure.

This is due to the fact that the β parameter value for

the membrane is very close to 1, and for this value,

preventive replacements are unproﬁtable at all (Naka-

gawa, 2003).

Figure 6: Membrane Weibull PDF for α = 51.559 and β =

1.021.

3.2 Swivel Joint on an Industrial Robot

The swivel joint is an element installed on the head

of an industrial robot that applies a seal to the

car window. It connects the material feeding hose

(polyurethane mass) with the movable robot head.

The tool on which the swivel is installed travels

Figure 7: Swivel joint Weibull probability plot determined

by coordinate linearization and less square method.

around the three edges of the glass approximately ev-

ery 70 seconds. During this time, the joint rotates 270

degrees and returns to the zero position at the end of

the robot cycle. Swivel joint wear occurs as a result

of repeated rotation, and its sign is loss of tightness or

blockage of movement. Damage of the swivel causes

the production line to stop completely for τ

= 1 h.

The cost of replacing it is C

M,2

= 60 EUR, and the

cost of stopping the line is the same as membrane, ap-

proximately C

D,2

= 650 EUR h. Therefore, the cost

of an emergency replacement for this item should be

ICEIS 2022 - 24th International Conference on Enterprise Information Systems

368

estimated at C

F,2

= C

M,2

+ τ

D,2

= 710 EUR. This

means the a factor for a swivel joint is 11.833.

Using the data on the failure rate of the swivel

joint, it is possible to estimate the parameters of the

distribution. The Figure 7 shows the process of deter-

mining the parameters using the least squares method.

The parameters calculated with this method have the

value α

= 210.551 and β

= 1.317. The Weibull dis-

tribution density plot for the determined parameters

is presented in Figure 8. Determining the optimal re-

Figure 8: Swivel joint Weibull PDF for α = 210.551 and

β = 1.317.

placement time for this item, we get T

= 87 days.

After this time, the swivel joint should therefore be

serviced.

3.3 Bando Belt

Bando belt is an element that transports glass waste

after the breaking operation. It works in a fast cy-

cle, every 16 seconds, starting and braking with great

torque. The belt works at ambient temperature and is

exposed to high tensile forces. The symptom of fail-

ure is partial or complete breaking of the belt. Dam-

age to the Bando belt causes a complete production

line stop at τ

= 4 h. The cost of replacing it is

M,3

= 1 000 EUR, and the cost of stopping the line

is the same as the memabrane and the swivel, approx-

imately C

D,3

= 650 EUR h. Therefore, the cost of an

emergency replacement of this element should be es-

timated at C

F,3

= C

M,3

+ τ

D,3

= 3 600 EUR. This

means that the a ratio for a Bando belt is 3.6.

Using the Bando belt failure rate data it is possi-

ble to estimate distribution parameters. The Figure

9 shows the process of determining the parameters

using the least squares method. The parameters ob-

tained with this method have the value α

= 54.849

and β

= 1.157. The Weibull distribution density plot

for the determined parameters is presented in Figure

10. Determining the optimal replacement time for this

Figure 9: Bando belt Weibull probability plot determined

by coordinate linearization and less square method.

Figure 10: Bando belt Weibull PDF for α = 54.849 and

β = 1.157.

item, we get T

= 237 days. After this time, the Bando

belt should be serviced.

3.4 IR Belt

The IR belt is a transport mesh that transports the

glass through the paint drying zone after the screen

printing process. The belt works with constant dy-

namics (speed and acceleration), but in an environ-

ment of increased temperature - 180 degrees Celsius.

Belt failure usually manifests as belt stretching be-

yond the adjustment value (critical length) or break-

ing completely. Damage to the IR belt causes the pro-

duction line to stop completely for τ

= 8 h. The cost

of replacing it is C

M,4

= 5 200 EUR, and the cost of

stopping the line is the same as for all other elements,

and is approximately C

D,4

= 650 EUR h. Therefore,

the cost of an emergency replacement of this ele-

ment should be estimated at C

F,4

= C

M,4

+ τ

D,4

Global Spare Parts Exploitation Costs Optymization and Its Reduction to Rectangular Knapsack Problem

369

10 400 EUR. This means that the a ratio for a Bando

belt is 2.0.

Figure 11: IR belt Weibull probability plot determined by

coordinate linearization and less square method.

Using the IR belt failure rate data it is possible

to estimate distribution parameters. The Figure 11

shows the process of determining the parameters us-

ing the least squares method. The parameters ob-

tained by this method have the value α

= 233.461

and β

= 2.264. The Weibull distribution density plot

for the determined parameters is presented in Figure

12. Determining the optimal replacement time for this

Figure 12: IR belt Weibull PDF for α = 233.461 and β =

2.264.

item, we get T

= 222 days. After this time, service

replacement of the IR belt should take place.

4 REDUCTION OF THE

OPTIMIZATION COST

PROBLEM FOR MULTIPLE

PARTS TO A RECTANGULAR

KNAPSACK PROBLEM

The examples presented in the previous section

clearly show that the optimal replacement times vary

greatly between the different parts, as it is shown in

Figure 13. Therefore, it becomes practically impossi-

Figure 13: Common Weibull PDF plot for all presented

parts.

ble to optimize the replacement time for all elements,

which may be even several hundred in the enterprise.

However, the question arises whether the downtime

caused by a failure cannot be used to replace other

parts as well, and not only the damaged one. Com-

puter aided calculations can estimate the proﬁtability

of immediate replacement of every spare part operat-

ing in the factory. Thus, the IT system may propose

preventive replacement of other parts to the mainte-

nance department at the time of failure. The effect of

using this type of approach will be the reduction of

the total emergency downtime of the line, which will

contribute to the reduction of maintenance costs in the

enterprise.

The presented proposal is a deﬁnitely new ap-

proach to the problem of optimization of operating

costs in an enterprise. The classical methods are lim-

ited to the use of analytical methods of the probability

theory and reliability theory. This article presents the

global optimization method, which combines analyti-

cal and discrete methods. The cost reduction method

presented here is one of many possible approaches

to optimizing the performance of maintenance teams.

However, it ﬁts perfectly into modern trends in predic-

tive maintenance for the needs of large-scale produc-

tion of automotive industry companies. It also opens

up wide possibilities of using machine learning mech-

ICEIS 2022 - 24th International Conference on Enterprise Information Systems

370

anisms to better estimate the failure rate function F(t)

and to determine global minima for the operating cost

function.

In order to create an algorithm to support the

decision-making process in the maintenance system,

it is necessary to deﬁne a common measure that de-

termines the condition of the production line. Such

a measure will be the expected cost of keeping the line

operational from now until the next planned down-

time. Suppose the line uses n spares U

,...,U

Each of the U

parts has a set of parameters that de-

ﬁne it. It can therefore be said that

= (t

,α

,β

,µ

(t),C

M,i

D,i

,τ

,σ

), (7)

where t

is the current time of failure-free operation of

the element, α

,β

are the Weibull distribution param-

eters deﬁning the failure rate of F

and the reliability

of G

, µ

(t) is the expected uptime of the element after

working t time units, C

M,i

is the cost of component re-

placement during planned downtime, C

D,i

is the cost

of production line downtime, τ

is the downtime re-

quired to replace the component, and σ

is the part of

the human resources that is used during the replace-

ment of a given item. If it is assumed that the next

scheduled stoppage will take place for T units of time,

then the expected cost of operating a part of U

at that

time is

C(U

,T ) =

F,i

+ G

M,i

, (8)

where C

F,i

= C

M,i

+τ

D,i

is the cost of an emergency

replacement part,

= 1 + k

+ F

) is the expected

number of failures in time to the scheduled downtime,

and G

) is the probability of failure-free operation

of the part from the time of the last forecast failure

to the scheduled downtime. The expected number of

Figure 14: Expected failures in the case of not carry out

preventive replacement of the element.

failures can be obtained by assuming the following

values for k

and t



T − µ

)

(0)



, (9)

= min (T − µ

) − k

(0),T ) . (10)

Note that if T < µ

), then k

= −1 and the expected

number of failures is

= F

(T ). The interpretation of

and t

is shown in Figure 14.

Additionally, we deﬁne the alternative expected

cost of operating the U

part

,T ) = ¯s

F,i

+ (1 + G

))C

M,i

, (11)

for which ¯s

= s

+ F

), s

= bT /µ

(0)c and t

T − s

(0). This function expresses the expected al-

ternative costs that will be incurred in the event that

part of U

has been replaced at present on the occa-

sion of downtime due to failure of another part. The

interpretation of s

and t

is shown in Figure 15.

Figure 15: Expected failures in the event of a preventive

replacement of an element.

For the functions C and C

deﬁned in this way, we

create a sequence of values V

= C(U

,T ) −C

,T ),

which express the expected proﬁt on the replacement

of U

during the emergency downtime. If we assume

that the part with index j failed, then the optimization

problem can be reduced to a two-dimensional knap-

sack problem. It should be assumed that the knapsack

has the shape of a rectangle with a width of τ

and

a height of 1 − σ

, and the elements arranged in it are

rectangles of width τ

, height σ

and values V

, where

i 6= j. It should also be emphasized that the elements

placed in the knapsack cannot be twisted, as time and

human resources are not interchangeable. Figure 16

illustrates the problem described above.

Figure 16: Optimum use of downtime for the preventive

replacement of additional parts.

The literature devoted to this problem is very

rich. The best review article to date is (Neuenfeldt Jr.

et al., 2022). During a systematic review of bibli-

ographic items, the authors distinguished four basic

approaches to the construction of algorithms for solv-

ing the two-dimensional rectangular knapsack prob-

lem: approximation, exact, heuristic and hybrid. The

article presents a very in-depth bibliometric analysis

indicating which publications have the most signiﬁ-

Global Spare Parts Exploitation Costs Optymization and Its Reduction to Rectangular Knapsack Problem

371

cant impact on the development of algorithms solv-

ing this problem. The review includes over one hun-

dred publications issued by 2020. The intensity of

research in this area is also evidenced by numerous

publications issued in the last two years: (Zhou et al.,

2019), (Fırat and Alpaslan, 2020), (Wei et al., 2020),

(Li et al., 2021) and (Wang et al., 2021b).

5 CONCLUSIONS

The article presents how the methods of analytical

and discrete optimization can be used to minimize the

operating costs of a production line. The third part

presents analytical methods of cost optimization for

a single spare part. The MES (Manufacturing Ex-

ecution System) equipped with this type of mecha-

nisms can keep the maintenance manager informed

about the probability of failure of a given part and the

expected costs of its removal. It provides the cur-

rent knowledge of what events should be prepared

for and what to expect in the near future. On the

other hand, the equations (8) and (11) introduced in

the fourth part constitute the basis for estimating the

actual value of the service activity. The algorithm

of conduct presented in this section is to support the

decision-making process in the area of maintenance

from the perspective of the entire enterprise. The

complexity of the two-dimensional knapsack problem

indicates that the global optimization problem is ex-

tremely computationally difﬁcult. Therefore, the ac-

tual functioning of global methods of minimizing op-

erating costs in real time requires signiﬁcant comput-

ing power. Therefore, the presented approach requires

further research in the ﬁeld of, for example, simpliﬁ-

cation of the model, which will result in a reduction

of computational complexity.

Nevertheless, the use of heuristic methods, which

relatively quickly indicate the ﬁrst approximation of

the solution to the problem, will be appropriate in

this case. We should remember that the system is

supposed to support decision-making, and not com-

pletely eliminate the human factor. Therefore, even

suggestions of possible solutions will give very valu-

able information for the maintenance manager and

can be successfully used in modern MES systems.

REFERENCES

Aigner, R., Leitner, M., and Stoschka, M. (2018). Fatigue

strength characterization of al-si cast material incor-

porating statistical size effect. In Henaff, G., edi-

tor, 12th International Fatigue Congress (FATIGUE

2018), volume 165 of MATEC Web of Conferences.

Almeida, J. (1999). Application of weilbull statistics to the

failure of coatings. Journal of Materials Processing

Technology, 93:257–263.

Ansell, J., Bendell, A., and Humble, S. (1984). Age re-

placement under alternative cost criteria. Management

Science, 30(3):358–367.

Bai, Y.-L., Yan, Z.-W., Ozbakkaloglu, T., Han, Q., Dai,

J.-G., and Zhu, D.-J. (2020). Quasi-static and dy-

namic tensile properties of large-rupture-strain (LRS)

polyethylene terephthalate ﬁber bundle. Construction

and Building Materials, 232.

Barlow, R. and Proschan, F. (1965). Mathematical theory

of reliability. John Wiley & Sons, New York.

Barlow, R. and Proschan, F. (1975). Statisitcal Theory of

Reliability and Life Testing. Holt, Rinehart, Austin.

Berg, M. and Epstein, B. (1978). Comparison of age, block,

and failure replacement policies. IEEE transactions

on Reliability, 27(1):25–29.

Chmielowiec, A. and Klich, L. (2021). Application of

python libraries for variance, normal distribution and

Weibull distribution analysis in diagnosing and op-

erating production systems. Diagnostyka, 22(4):89–

105.

Chmielowiec, A., Wo

s, W., and Gumieniak, J. (2021). Vis-

cosity Approximation of PDMS Using Weibull Func-

tion. Materials, 14(20):6060.

Christer, A. (1978). Reﬁned asymptotic costs for renewal

reward processes. Journal of the Operational Re-

search Society, 29(6):577–583.

eroux, R., Dubuc, S., and Tilquin, C. (1979). The age

replacement problem with minimal repair and random

repair costs. Operations Research, 27(6):1158–1167.

eroux, R. and Hanscom, M. (1974). Age Replacement

With Adjustment and Depreciation Costs and Interest

Charge. Technometrics, 16(2):235–239.

Finkelstein, M., Cha, J., and Levitin, G. (2020). On

a new age-replacement policy for items with observed

stochastic degradation. Quality and Reliability Engi-

neering International, 36(3):1132–1143.

Fırat, H. and Alpaslan, N. (2020). An effective approach to

the two-dimensional rectangular packing problem in

the manufacturing industry. Computers & Industrial

Engineering, 148:106687.

Fok, S., Mitchell, B., Smart, J., and Marsden, B. (2001).

A numerical study on the application of the weibull

theory to brittle materials. Engineering Fracture Me-

chanics, 68:1171–1179.

Fox, B. (1966). Letter to the Editor – Age Replacement with

Discounting. Operations Research, 14(3):533–537.

Frees, E. and Ruppert, D. (1985). Sequential nonparamet-

ric age replacement policies. The Annals of Statistics,

13(2):650–662.

Fu, Y., Yuan, T., and Zhu, X. (2018). Optimum pe-

riodic component reallocation and system replace-

ment maintenance. IEEE transactions on reliability,

68(2):753–763.

Glasser, G. (1967). The age replacement problem. Techno-

metrics, 9(1):83–91.

ICEIS 2022 - 24th International Conference on Enterprise Information Systems

372

Hemphill, M., Yuan, T., Wang, G., Yeh, J., Tsai, C.,

Chuang, A., and Liaw, P. (2012). Fatigue behavior of

al0.5cocrcufeni high entropy alloys. Acta Materialia,

60(16):5723–5734.

Ingram, C. and Scheaffer, R. (1976). On consistent esti-

mation of age replacement intervals. Technometrics,

18(2):213–219.

Keshevan, K., Sargent, G., and Conrad, H. (1980). Sta-

tistical analysis of the hertzian fracture of pyrex glass

using the weibull distribution function. Journal of Ma-

terials Science, 15:839–844.

Knopik, L. and Migawa, K. (2017). Optimal age-

replacement policy for non-repairable technical ob-

jects with warranty. Eksploatacja i Niezawodnosc -

Maintenance and Reliability, 19(2):172–178.

Lai, C. (2014). Generalized Weibull Distributions. Springer,

Heidelberg.

Lai, C., Murthy, D., and Xie, M. (2006). Weibull distri-

butions and their applications. In Pham, H., editor,

Springer Handbook of Engineering Statistics, pages

63–78. Springer London, London.

eger, C. and Cl

eroux, R. (1992). Nonparametric age re-

placement: bootstrap conﬁdence intervals for the op-

timal cost. Operations research, 40(6):1062–1073.

Li, Q., Fang, J., Liu, D., and Tang, J. (2003). Failure proba-

bility prediction of concrete components. Cement and

Concrete Research, 33:1631–1636.

Li, Y.-B., Sang, H.-B., Xiong, X., and Li, Y.-R. (2021).

An improved adaptive genetic algorithm for two-

dimensional rectangular packing problem. Applied

Sciences, 11(1):413.

Nakagawa, T. (2003). Maintenance and optimum policy. In

Pham, H., editor, Handbook of reliability engineering,

pages 367–395. Springer.

Nakagawa, T. (2006). Statistical models on maintenance. In

Pham, H., editor, Springer Handbook of Engineering

Statistics, pages 835–848. Springer London, London.

Neuenfeldt Jr., A., Silva, E., Francescatto, M., Rosa, C.,

and Siluk, J. (2022). The rectangular two-dimensional

strip packing problem real-life practical constraints:

A bibliometric overview. Computers & Operations

Research, 137:105521.

Newell, J., Kurzeja, T., Spence, M., and Lynch, M. (2002).

Analysis of recoil compressive failure in high perfor-

mance polymers using two-, four-parameter weibull

models. High Performance Polymers, 14:425–434.

Queeshi, F. and Sheikh, A. (1997). Probabilistic characteri-

zation of adhesive wear in metals. IEEE Transactions

on Reliability, 46:38–44.

Ran, A. and Rosenlund, S. (1976). Age replacement with

discounting for a continuous maintenance cost model.

Technometrics, 18(4):459–465.

Scmeaffer, R. (1971). Optimum Age Replacement Poli-

cies with an Ikreasing Cost Factor. Technometrics,

13(1):139–144.

Sheikh, A., Boah, J., and Hansen, D. (1990). Statistical

modelling of pitting corrosion and pipeline reliability.

Corrosion, 46:190–196.

Subramanian, R. and Wolff, M. (1976). Age replacement in

simple systems with increasing loss functions. IEEE

Transactions on Reliability, 25(1):32–34.

Wang, J., Qiu, Q., and Wang, H. (2021a). Joint optimization

of condition-based and age-based replacement policy

and inventory policy for a two-unit series system. Re-

liability Engineering and System Safety, 205:107251.

Wang, T., Hu, Q., and Lim, A. (2021b). An exact algorithm

for two-dimensional vector packing problem with vol-

umetric weight and general costs. European Journal

of Operational Research.

Wei, L., Lai, M., Lim, A., and Hu, Q. (2020). A branch-and-

price algorithm for the two-dimensional vector pack-

ing problem. European Journal of Operational Re-

search, 281(1):25–35.

Weibull, W. (1939). A statistical theory of the strength of

material. Ingeniors Vetenskaps Akademiens Handli-

gar, 151:5–45.

Xie, S., Lin, H., Wang, Y., Chen, Y., Xiong, W., Zhao,

Y., and Du, S. (2020). A statistical damage consti-

tutive model considering whole joint shear deforma-

tion. International Journal of Damage Mechanics,

29(6):988–1008.

Zhao, X., Al-Khalifa, K., Hamouda, A., and Nakagawa, T.

(2017). Age replacement models: A summary with

new perspectives and methods. Reliability Engineer-

ing & System Safety, 161:95–105.

Zhao, X., Li, B., Mizutani, S., and Nakagawa, T. (2021).

A Revisit of Age-Based Replacement Models With

Exponential Failure Distributions. IEEE Transactions

on Reliability.

Zhou, S., Li, X., Zhang, K., and Du, N. (2019). Two-

dimensional knapsack-block packing problem. Ap-

plied Mathematical Modelling, 73:1–18.

Global Spare Parts Exploitation Costs Optymization and Its Reduction to Rectangular Knapsack Problem

373