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Engineering, Technology & Applied Science Research Vol. 12, No. 3, 2022, 8548-8554 8548 
 

www.etasr.com Bella & Zohra: Application of Fminsearch Optimization to Minimize Total Maintenance Cost with the … 

 

Application of Fminsearch Optimization to Minimize 

Total Maintenance Cost with the Aim of Reducing 

Environmental Degradation 
 

Yasmina Bella 

LAS Laboratory 

Electrical Engineering Department 
Ferhat Abbas Sétif University 1 

Setif, Algeria 

b_yasmina2014@ univ-setif.dz 

Kebbab Fatima Zohra 

DACHR Electrical Engineering Department 
Ferhat Abbas Sétif University 1 

Setif, Algeria 

fatimazohra.kebbab@univ-setif.dz 

 

Received: 7 March 2022 | Revised: 24 March 2022 | Accepted: 28 March 2022 

 

Abstract-This study examined a production system under 

deterioration and its impact on the degradation of the 

environment. The environment degrades as the production 

system reaches a certain level of deterioration. To reduce 
environmental degradation, the system's deterioration is 

monitored through scheduled inspections, after which preventive 

or corrective actions are taken. To achieve the optimum 

inspection dates that minimize the average total cost per time 

unit, the Fminsearch algorithm was applied to calculate the 

optimal inspection dates for two cases of sequential inspection: 

periodic and aperiodic. To validate the performance of the 
proposed Fminsearch algorithm, simulation results were 

compared with the Nelder-Mead method. The comparison results 

showed the superiority of the Fminsearch algorithm in 

optimizing inspection maintenance dates to reduce the 
environmental degradation ratio. 

Keywords-environment degradation; fminsearch algorithm 
optimization; maintenance; Nelder-Mead; production system 

I. INTRODUCTION  

Environmental protection is described by international 
standards to which manufacturers must comply. The 
degradation of industrial systems can impact the environment 
in many ways. For example, the process of meeting energy 
demands raises concerns about energy sustainability and 
environmental protection, in conjunction with the market and 
regulatory requirements [1, 2]. Many environmental taxes have 
been imposed in recent decades all over the world. In the 
United States, the use of pollution prevention activities has 
increased significantly in the last two decades. The Pollution 
Prevention (P2) program is considered one of the main ways to 
reduce pollution [3, 4]. To ensure the efficient operation of an 
industrial system and meet the requirements of the 
environmental protection standards, inspection and 
maintenance activities are carried out. Inspection aids in 
controlling the degradation process of a production system and 
collecting crucial data on its reliability to determine the 
maintenance measures to be taken. Since the difficulties faced 
in maintenance inspections are attracting a lot of attention in 

the industry, many inspection strategies have been developed. 
In [5], the optimal inspection dates and treats for the sequential 
inspection strategy were calculated, while a similar model was 
used in [6]. In [7], the same model was proposed with the 
estimate of the delay of the inspections, integrating the penalty 
cost in the mathematical model. The optimal inspection period 
and the maintenance threshold of system degradation were 
calculated in [8, 9]. A mathematical model was developed in 
[10] for the overall cost, including the production cost along 
with the maintenance action cost. 

This study analyzed an inspection strategy that covers the 
environmental impact of the degradation of a production 
system. This strategy decreases degradation through 
maintenance actions to reduce environmental impact [11-13]. 
The system under consideration is submissive to continuous 
growing degradation. In this system, failure is detected only 
when it occurs, while the level of degradation of the system is 
only known after periodic or aperiodic sequential inspections. 
After checking the level of degradation when it exceeds a 
threshold, preventive actions are planned after a fixed 
timeframe, while corrective actions are carried out immediately 
after a system failure [14]. The main task was to minimize 
environmental degradation by decreasing the overall 
maintenance cost. The total cost is made up of the cost of 
corrective and preventive operations and inspection actions and 
the cost of penalties due to the environmental impact of system 
degradation. The main contribution of this study is the 
resolving of the optimization problem based on the Fminsearch 
algorithm. This algorithm is a direct search method to reduce 
non-linear functions [15-17]. 

II. DESCRIPTION OF THE PROBLEM  

A. Notations 

The following notations are used: 

• ���: The average cost of corrective maintenance. 
• ���: The average cost per time unit of severe environmental 

degradation. 

Corresponding author: Yasmina Bella



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www.etasr.com Bella & Zohra: Application of Fminsearch Optimization to Minimize Total Maintenance Cost with the … 

 

• ����: The average cost of an inspection. 
• �	�: The average cost of preventive maintenance. 
• 
�	: The timeframe to perform preventive maintenance. 
• Tth: A random elapsed time from when the system 

degradation began until it reaches the threshold. 

• �: Variable of 

� on the time axis. 
• ��: The probability density function of	

�. 
• � : A random elapsed time from instant �  until failure 

(lifetime of the system after exceeding the threshold). 

• �: Variable of �, from the instant �. 
• ��, ��: probability density and distribution function of � 

respectively. 

• ����: A random number of examinations during a cycle.  
• �∗���,��,…���: The vector of examination dates. 
• Trd: A random time of excessive environmental 

degradation. 

• Tcy: A random cycle time measured between two 
consecutive maintenance actions: preventive or corrective. 

• ��� : Probability that the cycle will end with corrective 
maintenance action. 

• �	� : Probability that the cycle will end with preventive 
maintenance action. 

• Ctot: The total cost incurred by maintenance and 
examination operations and by environmental degradation 
during a cycle. 

B. Assumptions 

This study was based on the following assumptions: 

• After each inspection, two types of action are likely: to 
leave it as it is or perform preventive maintenance. 

• Corrective maintenance is performed immediately after a 
system failure. 

• The inspection actions are assumed to be perfect, i.e. the 
inspection reveals the actual level of the system’s 
degradation without error. 

• Corrective and preventive maintenance actions are meant to 
be perfect. 

• The durations of inspection and maintenance actions are 
negligible. 

• System degradation leads to environmental degradation. 

• The system only fails if its degradation level exceeds the 
alarm threshold. Such a failure is supposed to be detected 
automatically (self-declared failure case). 

• The system works as it should after corrective or preventive 
maintenance actions. 

• The costs		���,	�	�	, ���� and	���, with the timeframe 
�	 
as well as the densities �� and �� are kept in the system’s 
database. 

• Preventive maintenance action is scheduled after a 
timeframe 
�	, if the inspection reveals that the level of 
degradation has exceeded the alarm threshold, as seen in 
Figure 1. Any inspections in this interval are canceled. As 

an example: in the interval [��, �� +
�	], the inspections 
are canceled. 

 

 
Fig. 1.  Evolution of system degradation and distribution of examinations 
over time. 

III. THE MATHEMATICAL MODEL OF MAINTENANCE COST 

This section presents the mathematical model of the 
average total cost per time unit Q of maintenance. Q is defined 
as the ratio between the average global cost  ��
!
� and the 
average cycle time  �
�"� [18]: 

#	 $ 	 ��
!
�	/	 �
�"�     (1) 
This cost includes the cost of preventive and corrective 

actions, the inspection cost, and the penalty cost. In the case of 
corrective action, the average cost is ���� &	����, while the 
expression 		��	� & �	�� indicates the average cost in the case 
of preventive maintenance.  ������	is defined as the average 
number of inspections during the time cycle, and their average 
cost is given by 	���� 	& 	 ������.  �
��	� is the average time 
of the critical environmental degradation and the expression ��� & 	 �
��	� corresponds to the penalty cost due to severe 
environmental degradation. The following formula illustrates 
the average cost: 

 ��
!
� $	���� &	���	�	' ��	� & �	��		' ����� 	&	 ������� '	���� & 	 �
����    (2) 
This formula uses the proposals developed in [4, 18]. The 

probability ���  that a time cycle will end with corrective 
actions is given by:  

	��� $ ∑ ) ����� 	' 
�	 	* ��	�����+�,-,-./��0�     (3) 
The probability that corresponds to the case where the time 

cycle ends with preventive action is given by: 

�	� 	$ 1 * ���    (4) 
The average number  ������ of examinations during a time 

cycle is given by: 

 ������ $ ∑ 2	���0� ) ����3� * ��	�����+�	,-4/5 × 
) ���� 	* 	��	�����+��,-	5     (5) 



Engineering, Technology & Applied Science Research Vol. 12, No. 3, 2022, 8548-8554 8550 
 

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The average time  �
6��  of excessive environmental 
degradation through a time cycle is given by: 

 �
6�� $ ∑ ) �) 71 * �����8+�������+�,-39:;<
5,-,-./��0�     (6) 
The expression of the numerator of the function presented 

by (1) is concluded from (2) - (6) as: 

 ��
!
� $ �=	��� 	*	�	�> ∑ ) ��=�� 	' 
�	 	* �>�����+�,-,-./��0� 	' �	� '���� ∑ 2	���0� ) ����3� * �������+�	,-4/5 ) ���� 	* 	��	�����+��,-	5 '
��� ∑ ) �) 71 * �����8+��	�����+�			,-39:;<
	5,-,-./��0�     (7) 

The denominator of (1) is given by:  

 �
�"� $ ∑ ) �	?-?-./��0� t ' ) 71 * �����8	+�	�
?-39:;<A
5 ���t�+t  (8) 

From (7) and (8), the general expression of the 
mathematical model of the average total cost per time unit of 
the maintenance function defined by (1) is deduced as: 

#���� $

BC
CC
D �=	EFG	<	E;G>∑ ) HI=,-	39:;	<
>JI�
��
K-K-./

L-M/ 	3E;G											
3N-LO ∑ �	�L-M/ ) H�,-4/<
�JI�
��
	K-4/P ) H�,-	<	
�	JI�
��
�K-	P

3NQI ∑ ) �) 7�<HI�R�8�R�	JI�
��
			K-4ST;.U	PK-K-./
L-M/ VW

WW
X

∑ ) Y
3) 7�<HI�R�8�R
K-4ST;.U
P Z	JI�
��
K-K-./L-M/

  

(9) 

The minimum value of the average total cost per time unit 
minQ(φ) is considered an objective function. The Fminsearch 
optimization was applied to calculate the optimal inspections 
vector to reduce the environmental degradation ratio and 
minimize the maintenance cost value Q(φ). 

IV. OPTIMIZATION PROCEDURE 

A periodic and an aperiodic sequential examination plan 
were applied to solve the problem without any constraint on the 
examination periods. In the aperiodic plan, the interval between 
examinations is independent. This problem is a 
multidimensional nonlinear problem [19]. The mathematical 
model exposed in Section III was exploited to solve this 
problem. The objective was to find the optimal examination 
dates which minimize the average total cost per time unit Q(φi) 
given by (9). The Fminsearch algorithm was used to find the 
minimum of an unconstrained multivariable function using a 
derivative-free method. This algorithm has been used 
repeatedly to resolve nonlinear functions and is well known for 
its robustness and ease of implementation [14]. 

A. Analytical Procedure  

The flowchart shown in Figure 2 corresponds to the 
analytical approach and was developed to determine the 
optimal sequential examination dates that minimize the average 
global cost per time unit. At first, the input data:	��, ��, ���, �	� , ���, ���� and 
�	 are given. Afterward, the Fminsearch 
algorithm is applied to calculate the optimal vector of 
examination dates ( �∗���,��,…��� ). Then, the iteration 
number is tested: If the inequality n>N is unattained, n 
increases. If it was attained, iterations stop. Finally, the solution 
vector of examination dates �� that match up to the minimum 
value	#���� is selected. 

 

 

Fig. 2.  Pseudo code of the Fminsearch algorithm. 

Two techniques were proposed to calculate the optimal 
inspection dates of the production system: periodic and 
aperiodic sequential inspection. The results of the Fminsearch 
algorithm were compared to the Nelder-Mead optimization 
method [18, 20]. 

B. Numerical Applications  

The above procedure was used to estimate the optimal 
examination dates. The input data were provided: probability 

densities�� and��, the costs	���,	�	� ,���, ���� and time flow 
�	 . Time was expressed in time units, while cost was 
expressed in monetary units. λ is the ratio between the mean 
time  �
6��  of severe environmental degradation and the 
average time  �
�"	� of the time cycle: 

[ $ \�9]I�\�9F^�    (10) 
Therefore: 

[ $ ∑ ) Y) 7�<HI�R�8�R
K-4ST;.UP ZK-K-./ 	JI�
��

L-M/
∑ ) Y
3) 7�<HI�R�8�R

K-4ST;.U
P Z	JI�
��
			K-K-./L-M/

    (11) 

1) Periodic Sequential Examination 

In this case, inspection is performed periodically. The 
inspection period ∆φ is defined as:  

∆	� $	���� * ���<�� $ ��`		,						2	 $ 	1,2	..n    (12) 
The probability densities �� and �� correlate jointly to the 

random variables 
�" and X. The probability density �� is given 
by: 

��	�d� $ �eJ√�g `<
/
hij.klml n    (13) 

with µf = 1100 and o� = 150. The probability density		�� is 
given by:  

		���d� $ �ep√�g `
</hiq.krmr n

h
    (14) 

with µg = 130 and og = 40. The average costs are fixed as ��� = 16108, �	� = 988, and ���� = 150. ���  and 
�	  are 
introduced as parameters. 



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a) Application 1:  

To examine the impact of cost ��� over the cost #�	∆�∗� 
and the ratio 	[ , the analytical Fminsearch algorithm was 
implemented in Matlab. The delay 
�	  was considered 
negligible, so the preventive operations are executed 
immediately. The cost 	#�∆�∗� and the ratio [ are presented 
inTable I.  

TABLE I.  THE IMPACT OF COST ���OVER #�∆�∗� AND RATIO [ 
WITH 
�	 $ 0, IN THE CASE OF A PERIODIC INSPECTION STRATEGY 

tuv ∆w∗ x�yz{|� x�}~v� t��� x�}��� ��∆w∗� �7%8 
0 22 3.48 15.02 987.36 726 1.36 2.07 

10 19 2.01 8.50 985.53 541.5 1.82 1.57 

20 18 2.20 5.97 986.58 486 2.03 1.23 

30 15 3.54 3.51 985.50 337.5 2.92 1.04 

40 13 2.12 2.53 986.11 253.5 3.89 1.00 

50 12 1.62 1.96 987.12 216 4.57 0.91 
 

As shown in Table I, when 	���  increases, λ decreases, 
reducing the rate of environmental degradation.  �
���	decreases by increasing the penalty cost	���, therefore 
the rate of environmental deterioration decreases. Figure 3 
presents the variation of #�∆�∗� as a function of 	���	. The 
results obtained by the Fminsearch algorithm show a decrease 
in the value of #�∆�∗�. This value varies from 55.11% to 
2.76% compared to the Nelder-Mead method. 

 

 

Fig. 3.  Influence of penalty cost ��� over #�∆�∗� in the case of a periodic 
examination strategy. 

 
Fig. 4.  Impact of penalty cost ��� on the ratio [, in the case of periodic 
examination strategy. 

Figure 4 shows the variations of λ. The Fminsearch 
algorithm gives lower values that vary from 46.37% to 67.38% 

compared to the Nelder-Mead, for an increasing value of ���. 
On the other hand, the ratio λ decreases by increasing the 
penalty cost, i.e. the degradation of the system as well as that of 
the environment are reduced. 

b) Application 2  

The penalty cost ���  was fixed at 20 monetary units to 
study the impact of the duration 
�	 over #�∆�

∗ � and λ. Table 

II shows the optimal period between inspections ∆�∗  for 
different values of the parameter 
�	 , by applying the 

Fminsearch algorithm. From Table II, it is noted that for 
�	 = 

0, λ and #�∆�∗ � have a minimal value of 1.20% and 1.64, 
respectively. Therefore, preventive operations should be carried 
out as soon as possible, i.e. the delay 
�	, should be minimal. 

Figure 5 presents the variation of the optimal value of the cost 

#�∆�∗ � as a function of 
�	, The Fminsearch algorithm gives 

better cost optimization for low values of 
�	, (up to 56.50%). 

TABLE II.  THE IMPACT OF 
�	 OVER #�∆�
∗� AND [ WITH ��� $ 20, 

IN THE CASE OF A PERIODIC EXAMINATION STRATEGY. 

}��  ∆w
∗
 x�yz{|� x�}~v� t���  x�}��� ��∆w

∗ � �7%8 

0 20 2.41 7.2 984 600 1.64 1.20 

10 17 1.35 8.97 984.04 433.5 2.27 2.07 

20 14 2.75 9.73 987.84 294 3.36 3.31 

30 13 2.12 10.74 981.04 253.5 3.89 4.24 

40 11 1.23 8.67 987.36 181.5 5.44 4.78 

50 10 2.44 6.19 797.04 121.5 6.58 5.10 

 

 
Fig. 5.  Impact of timeframe	
�	 over	#�∆�∗�, in the case of periodic 
examination strategy. 

 

Fig. 6.  Influence of timeframe	
�	 over λ, in the case of a periodic 
sequential inspection strategy. 



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Figure 6 shows that the ratio λ increases with the 
lengthening of the time to carry out preventive operations. The 
Fminsearch algorithm gives an improved λ compared to the 
Nelder-Mead method. 

2) Aperiodic Sequential Examination 

The probability densities ��  and �� correspond to the 
random variables 

�  and X, respectively. It is assumed that 
each of the two densities follows the Weibull law whose scale 
parameter is expressed in time units, while the form parameter 
has no unit. The probability density �� is given by: 

�� ��� $ �
�l

�l
��



�l
��l<�	`<�

U
�l�

�l
    (15) 

where the scale parameter is �J= 1164 and the shape parameter 
is �J= 8.7. The probability density ��of the random time � 
elapsed from the instant � until the occurrence of the failure is 
given by: 

����� $ ��r�r��
R
�r��r<�	`

<� q�r�
�r
    (16) 

where the scale parameter is �p=144 and the form parameter is �p=3.6. The average costs are respectively fixed as ���=6180, �	�=4170, and ����=492, while ��� and 
�	 are introduced as 
parameters. 

a) Application 3:  

The effect of ��� on #��
∗) and [		was examined using the 

Fminsearch algorithm, when 
�	  is fixed at zero, i.e. when 

preventive maintenance actions are not delayed. Table III 
shows the optimal Q for each value of the optimal vector �∗. 
When ���= 0,  #�∆�

∗)  has a minimum and	[ has a maximum 
value of 3.23 and 3%, respectively. According to Table III, the 
increase in the ��� penalty cost induces an increase in the 
number of inspections. These additional inspections are 
necessary to detect, as quickly as possible, the exceeding of the 
degradation threshold to reduce the resulting cost, i.e. the 
reduction of (���×	 �
�+)). 

TABLE III.  THE IMPACT OF PENALTY COST ��� OVER # ��
∗) AND [, 

IN CASE OF A PERIODIC SEQUENTIAL INSPECTION STRATEGY\ 

tuv w
∗
 ��w∗) �% 

0 1.33, 5.99, 48.59 3.23 3.00 

10 2.18, 6.15, 10.32, 12.96, 54.21 3.48 2.67 

20 2.50, 5.89, 9.40, 15.04, 62.60 3.93 2.40 

30 1.32, 4.58, 8.74, 9.24, 23.99, 62.49 4.94 2.22 

40 1.32, 3.36, 8.74, 10.24, 24.03, 62.50 5.95 2.20 

50 1.40, 4.58, 8.60, 9.80, 25.30, 61.49 6.96 2.17 

 

Figure 7 shows the variation of the optimal value of the 
average total cost per time unit #�∆�∗)  according to the 
penalty cost ��� . The evolution of #�∆�

∗) is directly 
proportional to the ���  cost. The obtained results show that 
Fminsearch gives a better optimization of the cost #�∆�∗) 
compared to the Nelder-Mead algorithm, as there is a decrease 
in the value of #�∆�∗) up to 30.08%. The variation of the rate 
of degradation λ as a function of ��� is presented in Figure 8. 
The evolution of the ratio λ is inversely proportional to 	���, 
which implies that excessive environmental degradation is 

reduced. Figure 8 confirms the superiority of the Fminsearch 
algorithm compared to the Nelder-Mead in minimizing the 
degradation ratio. The Fminsearch algorithm has a maximum 
reduction of 60.05% compared to the Nelder-Mead at ���= 0 

 

 

Fig. 7.  Influence of the penalty cost		���on #��
∗) in the case of an 

aperiodic sequential inspection strategy. 

 
Fig. 8.  Influence of the penalty cost ��� on degradation rate λ in case of an 
aperiodic sequential inspection strategy. 

b) Application 4:  

To evaluate the impact of 
�	 over #��
∗)and λ, the cost 

���  was set to 30 currency units, while several values are 
assigned to the duration 
�	. Table IV shows the average total 

cost per time unit #��∗) with the corresponding ratio λ for 
each vector	�∗. 

TABLE IV.  INFLUENCE OF THE DURATION FP OVER #��
∗) AND [ IN 

CASE OF AN APERIODIC SEQUENTIAL EXAMINATION STRATEGY 

}�� w
∗ ��w∗) �% 

0 2.27, 3.73, 6.41, 7.79, 10.48, 63.39 4.00 2.59 

10 4.23, 6.35, 7.56, 19.00, 59.88 5.24 3.53 

20 3.43, 5.68, 21.51, 60.97 6.30 4.31 

30 3.70, 5.76, 19.18, 56.98 7.32 5.14 

40 1.28, 8.37, 54.14 8.38 5.92 

50 1.24, 8.79, 55.29 9.36 6.26 

 

According to Table IV, in the case where the duration 
�	 
is set to zero, the ratio λ is minimal and equal to 2.59%, and the 
optimal value of the average total cost per time unit #��∗) is 
4.00. Figure 9 shows the optimal value of the average total cost 
per time unit #��∗)  concerning 	
�	 . The Fminsearch 

[%
]



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algorithm gives a better optimization of the cost, up to 53.04%, 
for low values of		
�	. Figure 10 shows the degradation rate λ 

as a function of 
�	. The Fminsearch algorithm improves the λ 

up to 53.99% compared to the Nelder-Mead algorithm. 

 

 
Fig. 9.  Influence of the duration 
�	over #��

∗) in the case of an aperiodic 

sequential inspection strategy. 

 

Fig. 10.  Influence of the duration 
�	  on [  in the case of an aperiodic 

sequential inspection strategy. 

V. CONCLUSION  

This study aimed to maintain a production system whose 
declining performance causes environmental degradation by 
applying the Fminsearch algorithm to optimize periodic and 
aperiodic inspection maintenance dates. This method improved 
the impact of the environmental degradation cost ��� and the 
duration 
�	  on the total average maintenance cost and the 

ratio of environmental degradation. Moreover, the results of 
this method were compared to those of the Nelder-Mead 
method. The simulation results showed the superior 
performance of the Fminsearch algorithm in the optimization 
of inspection maintenance dates to reduce the ratio of 
environmental degradation. 

Future work could examine the modification of certain 
assumptions, such as the duration of inspections and 
maintenance actions. These hypotheses could enable the study 
of the production system availability. Other optimization 
methods such as metaheuristics or simulation methods could be 
applied, while the industrial production could be described with 
real data. 

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