International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 1

ISSN 2229-5518

FUZZY PROGRAMMING APPROACH FOR A COMPROMISE ALLOCATION OF REPAIRABLE COMPONENTS

Irfan Ali* and S. Suhaib Hasan

ABSTRACT

The present paper, considered the allocation problem of repairable components for a parallel-series system as a multi-objective optimization problem for two different models. In the first model, the reliability of subsystems are considered as different objectives. While in the second model, the cost and time spent on repairing the components are considered as two different objectives. Selective maintenance operation is used to select the repairable components and a fuzzy programming algorithm is used to obtain compromise allocation of repairable components for the two models under some given constraints. A numerical example is also given to illustrate the procedure.

Key Words: Reliability, Fuzzy Programming, Compromise allocation, Selective Maintenance, Multi-objective programming.

-----------------------*---------------------

1. INTRODUCTION

1  (1  r ) ni ai  di

(2)
In every industry, systems are used in the production of
goods. If such systems deteriorate or fail, then effect can

R =  i

i 1

be wide spread. System deterioration is often reflected in higher production cost, time lower product quality and quantity. The system maintenance decision is taken on
The repair time constraint for the system is given as

 ti d i  exp ( i d i )  T0

i 1

(3)
the basis of the state condition of the system (i.e. whether
where ti
is the time required to repair a component in
the system is good or bad). The aim is to present a model
of reliability improvement maintenance policies that

i  th subsystem and

exp (i

ai ) is the additional time

minimizes the total cost and time spent on maintaining a system. For this purpose, we consider a system which is a series arrangement of m subsystems and performing a
spent due to the interconnection between parallel
components (Wang et al. (2009)).
The repair cost constraint for the system is defined as

sequence of identical production runs.

c d

 exp ( d )  C

(4)
Suppose that after completion of a particular production

 i i

i1

i i 0

run, each component in the system is either functioning or failed. Ideally all the failed components in the subsystems are repaired and then replaced back prior to the beginning of the next production run. However, due to constraints on time and cost, it may not be possible to repair all the failed components in the system. In such
situation, a method is needed to decide which failed
where exp (i ai ) is the additional cost spent due to the interconnection between parallel components (Wang et al. (2009)).
However, in the event when the reliability of each
subsystems are of equally serious concern. Let us consider, for instance, the following multi-objective problem (see Ali et al. (2011c)):
components should be repaired and replaced back prior to the next production run and the rest be left in a failed

Maximize R1 , R2 , , Rm 

(i) 



condition. This process is referred to as selective maintenance (See Rice et al. 1998). In the selective maintenance the number of components available for
the next production run in the i th subsystem will be

subject

 ti d i

i1

 exp ( i d i )  T0



(ii )  (5)





(ni

 ai )  d i , i  1, 2, ..., m

(1)

 ci d i  exp (  i d i )  C0

i1

(iii ) 



The reliability of the given system is defined as

0  d i  ai , i  1,2,..., m and integers (iv) 

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 2

ISSN 2229-5518

where

R = 1  (1  r )

ni ai  di ,

i  1, 2, ..., m.

constrained optimization developed by LINDO Systems
Ali et al. (2011c) also discussed the situation in which time taken and the cost spent on system maintenance are minimized simultaneously for the required reliability R * (say). The mathematical model of the problem is defined as:

Min C   ci d i  exp  i d i  

i1 

Inc. A user’s guide- LINGO User’s Guide (2001) is also
available. For more information one can visit the site http://www.lindo.com.

2. THE FUZZY PROGRAMMING APPROACH

To solve multi-objective allocation problem of repairable components defined in equation (5), we apply the fuzzy programming approach (see Mahapatra et al. (2010)). At

and

 first, we find the upper bound

U r (best) and lower

m  bound



Lr (worst) for corresponding objective

Min T 

subject to

 ti d i  exp  i d i  

i1 



(6)

function Z r , r  1, 2,, m .

Let U r = aspiration level of achievement for objective r ,

Lr = lowest acceptable level of achievement for

1  1  r ni ai di

i1

 R* 



objective r ,

 r = U r  Lr = the degradation allowance for

0  d i

 ai

, i  1,2,..., m and integers 

objective r ,
The selective maintenance operation is an optimal decision-making activity for systems consisting of several components under limited maintenance duration. The main objective of the selective maintenance operation is to select the most important component in subsystem. Rice et al. (1998) were the first to deal with the selective maintenance problem. Ali et al. (2011a, 2011b, 2011c) and Faisal and Ali (2012) considered the problem of optimum allocation of repairable and replaceable components for series- parallel system by using selective maintenance.
Fuzzy programming offers a powerful means of
handling optimization problems with fuzzy parameters.
when the aspiration level and degradation allowance for
each objective are specified.

Algorithm:

Step 1: Solve multi-objective allocation problem of

repairable components as a single objective used each time and all other ignored.

Step 2: Determine the corresponding values for every

objective at each solution derived.

Step 3: Construct a pay-off matrix, according to every

objective w. r. to each solution the pay-off matrix in the main program gives the set of non dominated solution which shown in the following table.
Fuzzy programming has been used in different ways in the past. In (2010), Mahapatra et al. studied the fuzzy programming approach to stochastic transportation problem and many others. In reliability Park (1987),





d (1)



( 2)

Z11

Z 2

Z12

 Z m 



 Z1m 

Mahapatra and Roy (2006), Huang (1997), Dhingra
(1992), Rao and Dhingra (1992) and Ravi et al. (2000)

d Z 21



Z 22

 Z 2m 



have used fuzzy multi-objective optimization method to

  

   

solve reliability optimization problem having several conflicting objectives.

 ( m)



(1)

Z m1

( 2)

Z m2 

( m)

Z mm 

This paper presents a new contribution in the field of
where d

, d ,, d

is the ideal solution for the
system reliability optimization for compromise

objective Z1 , Z 2  Z m respectively.

allocation problem of repairable components. A Fuzzy
Let

Zij

 Zi (d

), i

 1, 2,,

m and

j  1, 2,, m

are
programming algorithm is used to obtain compromise
allocation of repairable components.
the maximum value (best) for each
The multi-objective NLPPS are solved by “Fuzzy programming algorithm” using software package LINGO. LINGO is a user’s friendly package for
objective Zi ,

i  1, 2,, m .

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 3

ISSN 2229-5518

Step 4: To find the best

(U r ) and worst (Lr ) for each

In a similar way, we solve multi-objective allocation
objectives corresponding to the set of solution i.e.

U r  Z rr and Li  MinZ1r , Z 2r ,, Z mr . To satisfy,

problem of repairable components defined in equation
(6); we apply the fuzzy programming approach. At first,

r 1

Z r  U r , r  1, 2,, m and constraints (ii), (iii) and (iv)

we find the upper bound

Lr (best) and lower bound

of NLPP (5).

Step 5: Construct membership function as,

U r (worst) for corresponding objective function

Z r where r  1, 2,, k .

Let Lr = aspiration level of achievement for objective r ,

0



 (d )   Z r  Lr

Zr ij   L

Z r  Lr

Lr  Z r  U r

U r = highest acceptable level of achievement for objective r ,

U r r

 r =

U r  Lr = the degradation allowance for

1

Z r  U r

objective r ,
when the aspiration level and degradation allowance for

If  Zr (dij )  1; then Z r is perfectly achieved,

each objective are specified.

 0; Z r

is nothing achieved,
Now construct membership function for model (2)

0  Zr (dij )  1; then Z r is partially achieved.

defined in equation (6) as,

Z  L

1 Z r  Lr

Step 6: Let  r

 r r , r  1, 2,, m

U r  Lr



 (d )  U r  Z r

Lr  Z r  U r

Using max-min/min-max operator, we have

maxmin(1 , 2 ,, m ),

U r  Lr

0

Z r  U r

then

we have; Max 

If  Zr

(dij )  1; then Z r is perfectly achieved,

1  

 0; Z r is nothing achieved,

 2  



0  Zr

(dij

)  1; then Z r is partially achieved.

U r  Z r

 m  

Let us define  r



U r  Lr

, r  1, 2,, k

 Z

(d ij );

 Next using max-min/min-max operator, we have

where   Max r

     

r i  j  1, 2,, m & r  1, 2,, m

max min( 1 ,

2 ,,

k ) ,

Finally we obtained the mathematical programming

then

we have; Max 

formulation for equation (5) through fuzzy programming , as follows:

1  

  

Maximize  





  

subject to

 k

 where

1  (1  r ) ni ai di   (U

 L )  L   

t d

i i i i



 exp ( d )  T 

(7)

  Max

 Z r

(d ij

); i  j  1, 2,, m & r  1, 2,, k

 i i

i 1

i i 0 



Finally we obtained the mathematical programming formulation for equation (6) through fuzzy

 ci d i  exp (  i d i )  C0 

i1 

  0 

0  d i  ai , i  1,2,..., m and integers 

programming as follows:

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 4

ISSN 2229-5518

Maximize 

 Maximize Ri



 1  (1  r ) ni ai di  



subject to



 subject to

i  1, 2, m 



 ti d i  exp (i d i )   (U1  L1 )  U1 

i 1 

m 



 ti d i  exp ( i d i )  60 

(9)



i 1

ci d i

 exp (  i

d i )   (U 2

 L2



)  U 2 



(8)

i 1

 c d

 exp ( 



d )  90 

1  1  r ni ai di



 R* 

i i i i i 1



i 1

  0

 1  d i  ai , i  1,2,..., 5 and integers 



0  d i  ai , i  1,2,..., m and integers 

3. NUMERICAL ILLUSTRATION

Consider a system consisting of 3 subsystems. The
Using the equation (9) construct a pay-off matrix,
according to every objective with respect to each solution the pay-off matrix in the main program gives the set of non dominated solution which shown in the following table
available time between two production runs for  Z Z
repairing and replacing back the components is 60 time  1 2
units. Let the given maintenance cost of the system be 90 

Z 3 



units. The other parameters for the various subsystems
are given in table 1.

Table 1: The parameters for the numerical example

d (1)



d ( 2)



0.9992433

0.9589938

0.9084938

0.9916266

0.9687500 



0.9687500 



d (3)

0.9589938

0.9084938

0.9990234 

The upper bounds for the given model 1 are

U1  0.9992433,U 2

 0.9916266,U 3

 0.9990234 a

nd lower bound

L1  0.9589938, L2  9084938, L3  0.9687500 .

Using the equation (7), we can formulated the model 1as

Solution by Using Fuzzy Programming Approach

Model 1: Using the values given in Table 1 the NLPP (5)

Maximize  



Subject to 

and their optimal solutions

d (i ) ; i  1, 2 and 3

with 

(3d )  

1  (1  0.55)

1  0.0402495  0.95899 

the corresponding values of 
are listed below. These

 (3d2 )  

values are obtained by software LINGO.

1  (1  0.45)

 0.0831328  0.90849 

1  (1  0.55) (3di )  0.0302734  0.96875 



(10)

 ti d i  exp ( i d i )  60 

i 1 

m 

 ci d i  exp ( i d i )  130 

i 1 

  0 



1  d i  ai , i  1,2,..., m and int egers 

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 5

ISSN 2229-5518

The above problem (10) is solved by the LINGO
Software for obtaining the optimal solution of the
practical utility of the fuzzy programming approach in
system reliability.
problem. We get

  0.7741945

and the compromise

  

solution as d1

 3, d 2  4, d 3  3 . The optimal

REFERENCES

reliabilities of each subsystem are

R  0.9916962, R  0.9847756, R  0.9921875 .

1 2 3

Model 2: Using the values given in Table 1 the NLPP (6)

[1]. Ali, I., Raghav, Y. S., and Bari, A., (2011); Allocating

Repairable and Replaceable Components for a System availability using Selective Maintenance: An Integer Solution; Safety and Reliability society, 31(2), 9-18.
for the desired reliability requirement

R*  0.97

has

[2]. Ali, I., Khan, M. F., Raghav, Y.S., and Bari, A., (2011).

been solved and construct a pay-off matrix, according to
every objective with respect to each solution the pay-off
matrix in the main program gives the set of non dominated solution which shown in the following table for
Allocating Repairable and Replaceable Components for a System Availability using Selective Maintenance with probabilistic constraints; American Journal of Operations Research, 1(3):147-154.

[3]. Ali, I., Raghav, Y.S., and Bari, A., (2011c); Integer





d (1)



d (2)

62.38

64.66

Z 2 



141.72 



141.30 

goal programming approach for finding a compromise allocation of repairable components, International Journal of Engineering Science and Technology, 3(6): 184-195.

[4]. Dhingra, A.K. (1992); Optimal apportionment of reliability and redundancy in series systems under multiple objectives, IEEE Trans. Rel., 41, 576-582.

The upper bound and lower bond for the model 2 are

U1  64.66, U 2  141.72

and L1  62.38, L2  141.30 .

[5]. Huang, H.Z. (1997); Fuzzy multi-objective optimization decision-making of reliability of series system, Microelectronics Reliability, 37 (3), 447–449.

[6]. Khan, M. F. and Ali, I. (2012); Allocation Problem of

Maximize 

subject to

 Repairable and Replaceable Components for a System

 Reliability using Selective Maintenance: An Integer



 Solution, International Journal of Scientific and Engineering

 ti d i  exp (i d i )  2.38  64.66 

i1 

m 

Research, 3(5), 1-4.

[7]. LINGO User’s Guide (2001), published by Lindo

Systems Inc., 1415 North Dayton Street, Chicago,

c d

 exp (

d )  0.42  141.30 

(11)
Illinois-60622 (USA).

 i i i i

i1



 [8]. Mahapatra, D. R., Roy, S. K. and Biswal, M. P. (2010);

 Stochastic Based on Multi-Objective Transportation

1  1  r ni ai di

i 1

 0.97 



Problems Involving Normal Randomness, Advance

  0 

1  d i  ai , i  1,2,..., m and integers 

The above problem (11) is solved by the LINGO Software for obtaining the optimal solution of the
Modeling and Optimization, 12 (2), 205-223.

[9]. Mahapatra, G. S. and Roy, T. K. (2006); Fuzzy Multi-

Objective Mathematical Programming on Reliability Optimization Model, Applied Mathematics and Computation, 174 (1), 643-659.

[10]. Park, K. S. (1987); Fuzzy apportionment of system

problem. We get

  0.0233

and the compromise
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  

solution as d1

 3, d 2  4, d 3  4 .

[11]. Rice, W.F., Cassady, C.R., and Nachlas, J.A., (1998);

Optimal Maintenance Plans under Limited Maintenance

6. CONCLUSION

This paper is an attempt to utilize Fuzzy programming approach to the solution of optimum compromise allocation of repairable components in a system. Further, a numerical example is presented to demonstrate the
Time, Industrial Engineering Research 98 Conference proceedings.

[12]. Rao, S. S. and Dhingra, A. K. (1992); Reliability and

redundancy apportionment using crisp and fuzzy multi- objective Optimization approaches, Reliability Eng. Syst. Safety, 37, 253-261.

International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 6

ISSN 2229-5518

[13]. Ravi, V., Reddy, P. J. and Hans-Jurgen

Zimmermann (2000); Fuzzy global optimization of complex system reliability, IEEE Trans. Rel., 8, 241-248.

[14]. Wang, Z., Chen, T., Tang, K. and Yao, X. (2009); A

Multi-objective Approach to Redundancy Allocation Problem in Parallel- series System, IEEE Congress Evolutionary Computation, 582-589.

BIOGRAPHICAL NOTES

 Irfan Ali received M.Sc. and M. Phil. in Statistics from Aligarh Muslim University, Aligarh, India. Presently he is pursuing Ph.D. from department of Statistics and Operations research Aligarh Muslim University, Al igarh, India. He has published 16 research papers in national and international journals of reputes. He has also presented 4 research articles in national conferences.
 Suhaib Hasan received M.Sc. and M. Phil. in Statistics from Aligarh Muslim University, Aligarh, India. He is a
Associate Professor in the Department of Statistics and Operations research Aligarh Muslim University, Aligarh, India. Mr. Hasan has more than 25 years of experience in teaching and research. His area of interest also includes linear models and econometrics.