Algorithm of Maintenance Time and Maintenance Amount Based on Maintenance Degree

Xiaobo Su1, 2, Qi Gao1, Weining Ma1, Yukun Chen1 1Shijiazhuang Division of Army Engineering University, Shijiazhuang China

2Shijiazhuang Division of PLAA Infantry Academy, Shijiazhuang China

Keywords: algorithm, maintenance degree, maintenance time, maintenance amount

Abstract: Under the premise that available time and maintenance degree is given, using the relationship between maintenance degree and maintenance time, this paper set up one algorithm to judge if the maintenance mission can be accomplished in certain time and put forward one method to calculate the equipment amount be repaired in available time, which are extremely helpful in the process of maintenance decision especially in the support of computer.

1. Introduction

In recent years many new equipment are equipped to army. Because of the increased training intensity and there are few experience to reference about how to use these new equipment correctly, the amount of equipment damage is huge. New equipment is often made up of a lot of complex units which lead to complex damage model. So the maintenance tasks are very heavy. On the other hand the maintenance time can be used is shortage in war. Under these circumstances there are 2 problems is concerned by the commander. One is how long the damaged equipment can be repaired. The other is how many damaged equipment can be repaired in certain time. The two problems is closely related to time. There are many study on method to calculate maintenance time. Liu Duan and Hu Jianbo make use of Monte Carlo method to set up one model to calculate the maintenance time [1]. Zhang Lijun, Liu Yi, and Liu Jia put forward one maintenance time prediction method based on virtual reality [2].Shen Guixiang, Zeng Wenbin put forward one method to determine the system’s average maintenance time based on the units’ failure ratio and maintenance time model [3]. All these methods are under the conditions that there are enough time to repair the damaged equipment completely. They are suit for maintenance in peace time. But in wartime time can be used for maintenance is limited and the equipment demand is urgent. Around these questions this paper study the issues on how to determine maintenance time and how many equipment can be repaired in certain time. These study are helpful for the commander to make decision.

2. Modelling Analysis

2.1 Analysis on Maintenance Degree Demand

In war the task is often urgent, the fight is often fierce, the time is short and equipment

2018 3rd International Conference on Computer Science and Information Engineering (ICCSIE 2018)

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maintenance demand is heavy. On the other hand, to repair damaged equipment thoroughly often need so long time as to miss the good opportunity to fight [4]. So in war the maintenance demand is often partly repair the damaged equipment especially the part of main weapons. And the maintenance degree is often less than 100 percent.

2.2 Analysis on Maintenance Time

In wartime under the assumption the maintenance power is certain, the maintenance time is restrict by two factors. One is maintenance degree. As the maintenance degree demanded is even higher, the maintenance time needed is even longer. The other is combat operations arrangement. The maintenance time can be used is often restricted by the combat operation arrangement. For example the maintenance work often be carried out in the rest time of the army march. That to say the maintenance time is the rest time. Another example is maintenance often be carried out in the break of one combat so that these repaired equipment can attend the next combat. The maintenance time is the combat break time. If the maintenance mission have not be finished in the break as to the equipment could not attend the next combat, it can be consider the maintenance would lose its effect.

3. Maintenance Time Model

3.1 Symbol Setting

m ――The amount of maintenance units; n ――Maintenance force kinds classified by technical level;

ijθ ――The maintenance time mean of the i unit is repair by the j kind of maintenance force;

iλ ――Failure rate of the i unit; t ――Maintenance time;

0t ――Maintenance time can be used ( )M t ――Maintenance degree. ( )0n t ――The amount of equipment be repaired under the demand of maintenance degree.

N ――The amount of equipment need repaired.

3.2 Assumption and Basic Conditions

(1) The maintenance condition of different unit is the same. (2) The maintenance event of very unit of the system is independent. (3) The maintenance time of very unit follows normal distribution. (4) The demand of maintenance degree is ‘ ( )M t k≥ ’. (5) The maintenance time permit by the operation arrangement is ‘ 0t t≤ ’.

3.3 Model of Maintenance Degree

3.3.1 Feature Parameters of Maintenance Time

The mean and the variance is the basic parameters for normal distribution. Although the maintenance time is often one random variable, we can got the average time and the variance that every unit is repaired by every kinds of maintenance power. If the maintenance times is enough, the average time can be consider as the maintenance time mean. These are crucial theory to get the

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maintenance degree model. The average time that every unit is repaired by every kinds of maintenance power is given in

Table 1. Table 1 Mean of the maintenance time.

Maintenance

power kind 1

Maintenance power kind 2

… Maintenance

power kind n

Unit 1 11θ 12θ … 1nθ Unit 2 21θ 22θ … 2nθ

… … … … … Unit m 1mθ 2mθ … mnθ

The maintenance event of every unit is independent. According to the theory of normal distribution every unit’s maintenance time mean and maintenance time variance can be calculated.

1

1 n

i ijjn

θ θ=

= ∑ , ( )2

12

1

n

ij ij

idn

θ θ=

−=

−

∑

3.3.2 Maintenance Degree Function of the System

Because the maintenance event of every unit is independent and the maintenance time follows normal distribution, according to the theory of normal distribution the maintenance time of the system follows normal distribution. And the parameters of the system can be got and shown in Equation (1) and (2).

1 1

1 1 1

1 1 1

1 1

m n

i ijm m ni ji

i i ijm m mi i j

i i ii i i

n n

λ θλ

θ λθ θλ λ λ

= =

= = =

= = =

= = =⋅

∑ ∑∑ ∑ ∑

∑ ∑ ∑ (1)

( )

( )

2

2 1 112

1 1

1

2

1 1

21 1

1

1 11 1

11

m nn

i ijijm mi jj

i mi i

ii

m n

i ijm ni j

ij mi j

ii

dm m n n

n m

λ θθθ θ

λ

λ θθ

λ

= ==

= =

=

= =

= =

=

= − = −− − ⋅

= −−

∑ ∑∑∑ ∑

∑

∑ ∑∑ ∑

∑

(2)

Then the maintenance degree function can be given as Equation (3).

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( ) ( )

2

0

2

1 1

1

2

1 11 12

11 1

11

1 1( ) exp ( )

22

1exp

21 12 11

dt

m n

i iji j

m

ii

m nm n

i ijni ijm ni ji j

ij mij mji j

iiii

tM t t

dd

tn

n mn m

θ

p

λ θ

λ

λ θλ θθp θ

λλ

= =

=

= == =

== =

==

−= − ⋅

−

= −

−−−−

∫

∑ ∑

∑

∑ ∑∑ ∑∑∑ ∑

∑∑

20

1

dt

m

i

t

=

∫

∑

(3)

3.3.3 Model of Maintenance Time of the System

For the restriction of ‘ ( )M t k≥ ’ and function 3. Then

( ) ( )

2

1 1

1

202

1 11 12

1 11 1

11

1exp d

21 12 11

m n

i iji j

m

it i

m nm n

i ijm ni ijm ni ji j

ij mij mi ji j

iiii

tn

t

n mn m

λ θ

λ

λ θλ θθp θ

λλ

= =

=

= == =

= == =

==

−

−

−−−−

∑ ∑

∑∫

∑ ∑∑ ∑∑ ∑∑ ∑

∑∑

k≥

Let

( )

1 1

1

2

1 1

1 1

1

1

1 11

m n

i ijmi j

ii

m n

i ijm ni j

ij mi j

ii

tn

Z

n m

λ θλ

λ θθ

λ

= =

=

= =

= =

=

−

=

−−

∑ ∑∑

∑ ∑∑ ∑

∑

(4)

The maintenance degree function turn into one standard normal distribution function through substitute Z to the maintenance degree function 3. Then the maintenance degree demand changes as following.

From ( )M t k≥ to ( )Z kΦ ≥ . Through search the standard normal distribution table, the formula ‘ ( )kZ kΦ = ’ can be got. In

terms of the theory of normal distribution, the inequality ‘ kZ Z≥ ’ can be got. Substitute Z into the previous inequality, inequality 5 can be got Substitute Z from formula 4 into ‘ kZ Z≥ ’ then the inequality can be got and shown in Equation (5).

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( )

1 1

1

2

1 1

1 1

1

1

1 11

m n

i ijmi j

ii

km n

i ijm ni j

ij mi j

ii

tn

Z

n m

λ θλ

λ θθ

λ

= =

=

= =

= =

=

−

≥

−−

∑ ∑∑

∑ ∑∑ ∑

∑

(5)

From Equation (5), the maintenance time demand according to the demand of maintenance degree can be got and shown in Equation (6).

( )

2

1 1

1 1 1 1

1 1

1 1 11

m n

i ijm n m ni j

k ij i ijm mi j i j

i ii i

t Zn m n

λ θθ λ θ

λ λ

= =

= = = =

= =

≥ ⋅ − +−

∑ ∑∑ ∑ ∑ ∑

∑ ∑ (6)

Then according to the operation arrangement ‘ 0t t≤ ’, Equation (7) can be got.

( )

2

1 10

1 1 1 1

1 1

1 1 11

m n

i ijm n m ni j

k ij i ijm mi j i j

i ii i

Z t tn m n

λ θθ λ θ

λ λ

= =

= = = =

= =

⋅ − + ≤ ≤−

∑ ∑∑ ∑ ∑ ∑

∑ ∑ (7)

3.3.4 Analysis on Equipment Maintenance Planning

(1) ( )

2

1 10

1 1 1 1

1 1

1 1 11

m n

i ijm n m ni j

k ij i ijm mi j i j

i ii i

t Zn m n

λ θθ λ θ

λ λ

= =

= = = =

= =

≥ ⋅ − +−

∑ ∑∑ ∑ ∑ ∑

∑ ∑

Under the condition above, the damaged equipment can be repaired and can attend the next battle. In the planning the commander should consider these equipment be repaired. According to Equation (3), the maintenance degree can be calculated as Equation (8).

( ) ( )

2

1 1

1

0 22

1 11 12

1 11 1

11

1( ) exp

21 12 11

m n

i iji j

m

ii

m nm n

i ijm ni ijm n i ji jij mij m

i ji jii

ii

tn

M t

n mn m

λ θ

λ

λ θλ θθp θ

λλ

= =

=

= == =

= == =

==

−

= −

−− −−

∑ ∑

∑

∑ ∑∑ ∑∑ ∑∑ ∑

∑∑

0

0d

tt

∫ (8)

According to the statistics definition of maintenance degree, the amount of equipment repaired can be got approximately as Equation (9).

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( ) ( )

( ) ( )

0 0

2

1 1

1

22

1 11 12

1 11 1

11

1exp

21 12 11

m n

i iji j

m

ii

m nm n

i ijm ni ijm ni ji j

ij mij mi ji j

iiii

n t N M t

tn

N

n mn m

λ θ

λ

λ θλ θθp θ

λλ

= =

=

= == =

= == =

==

= ⋅

−

= ⋅ −

−−−−

∑ ∑

∑

∑ ∑∑ ∑∑ ∑∑ ∑

∑∑

0

0d

t

t

∫ (9)

(2) ( )

2

1 10

1 1 1 1

1 1

1 1 11

m n

i ijm n m ni j

k ij i ijm mi j i j

i ii i

t Zn m n

λ θθ λ θ

λ λ

= =

= = = =

= =

< ⋅ − +−

∑ ∑∑ ∑ ∑ ∑

∑ ∑

Under the condition above, the maintenance can’t meet the demand of maintenance degree. The policy maker should change the operation arrange or decline the demand of maintenance degree.

4. Example of Model Application

4.1 Problem Description

In one mission, according to the operation arrangement some force should have a rest about 20 minutes after reach some certain area and then to put into battle. The force have 100 weapons. After reach the certain area, there are 40 weapons are fault. The weapon is consist of 3 units. The first is control unit. The second is search unit. The third is fire unit. Questions are if the maintenance degree can reach to 0.9 before the repaired weapons be put into the coming battle and how many weapons can attend the coming battle.

Known conditions is given in Table 2. Table 2 Known conditions:

Raw hand Veteran artificer engineer Failure probability

iλ Mean repair time ijθ (minute) Control unit 18 15 13 10 0.3 Search unit 24 20 18 15 0.25

Fire unit 21 18 16 13 0.15

4.2 Steps of the Model Application

Step 1: To calculate the maintenance time mean of every unit

For 1

1 n

i ijjn

θ θ=

= ∑

Then

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Control unit: 1 11

1 18 15 13 1014

4

n

jjn

θ θ=

+ + += = =∑ minutes.

Search unit: 2 21

1 24 20 18 1519.3

4

n

jjn

θ θ=

+ + += = =∑ minutes.

Fire unit: 3 31

1 21 18 16 1317

4

n

jjn

θ θ=

+ + += = =∑ minutes.

Step 2: To calculate the maintenance time that meet the maintenance degree demand ‘0.9’ From ( ) 0.9kZΦ = to 1.28kZ = Substitute data in table 2 and 1.28kZ = into formula 6, the maintenance time (that) meet the

maintenance degree demand can be got ‘ 0.9 20.4t = ’ minutes. Step 3: To judge if the time is enough

0 20t = , 0.9 20.4t = , 0 0.9t t< Conclusion: the maintenance time can’t meet the maintenance degree demand. The policy maker

should increase the maintenance time or cut down the maintenance degree demand. Step 5: To calculate the amount of the weapon be repaired From Equation (8), ‘ ( ) ( )0 20 0.84M t M= = ’ can be got. From Equation (9), ‘ ( ) ( )0 0 40 0.84 33.6n t N M t= ⋅ = × = ’ can be got. Conclusion: The amount of the weapon be repaired is 33. That is to say after the maintenance in

the interval of rest, there are 93 weapons can attend the coming battle.

References

[1] Liu D., Hu J. B., et.(2014) Monte Carlo Simulation of Maintenance Time Based on System Maintenance Work Procedure. Fire Control & Command Control, 39(7), 119-123. [2] Zhang L. Y., Liu Y. and Liu J. (2016) Maintenance Time Prediction Using Virtual Reality. Journal of Computer-Aided Design & Computer Graphics, 28(8), 1383-1391. [3] Shen G. X., Zeng W. B., et.(2017) Determination of Average Maintenance Time of CNC Machine Tools Under Minimum Failure Rate. Journal of Jilin University (Engineering and Technology Edition), 47(5), 1519-1525. [4] Wang L. (2010) Research on Time Schedual Plan for battlefield Damage Assessment and Repair [D]. Master Thesis of National University of Defence Technology.

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