Behrouz Fathi Vajargah, Sara Ghasemalipour/ TJMCS Vol. 4 No.

3 (2012) 402 - 410

The Journal of
Mathematics and Computer Science

Available online at
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The Journal of Mathematics and Computer Science Vol. 4 No.3 (2012) 402 - 410

T-age Replacement Policy in Fuzzy Renewal Reward Processes

Behrouz Fathi Vajargah1,*, Sara Ghasemalipour2
1Department of Statistics, Faculty of Mathematical Sciences, University of Guilan,
Iran,[email protected],
2Department of Statistics, Faculty of Mathematical Sciences, University of Guilan, Iran,

[email protected]

Received: February 2012, Revised: May 2012

Online Publication: July 2012

Abstract
This paper studies a renewal reward process with fuzzy reward and fuzzy random inter arrival times. A
theorem about the long run average fuzzy reward and fuzzy life time is proved. The original problem is
evaluating the membership of the long run average fuzzy cost per unit time that for obtaining
membership, we should solve a nonlinear programming problem. Finally, some application example is
provided to illustrate the result.

Keywords: Fuzzy renewal reward processes, Fuzzy random reward, Fuzzy random variables,
Membership function, Fuzzy life time, Nonlinear programming

1. Introduction
Renewal reward processes are an important type of renewal models. The stochastic renewal reward theorem is
one of the most important results in this area. On the other hand, several researches recently studied on fuzzy
renewal processes. Zhao and Liu [1] discussed a fuzzy renewal process generated by a sequence of i.i.d positive
fuzzy variables obtained a fuzzy elementary renewal theorem and a fuzzy renewal reward theorem. Wang and
Watada [2] discuss a renewal reward process with fuzzy random interarrival times and rewards under the
independence with t-norms (T-independence), which induces the (generalized) t-norm-based extension principle
for the operations of fuzzy realizations of fuzzy random variables. There are many approaches taken in the
literature to decide on a maintenance strategy. For a system which consists of one component see Valdez-Florez
[3] for an extensive review. Different authors assume different cost structures. Usually the costs are either
constants or random variables. Constant costs will not contribute anything to modeling imprecision in the

1,*
Corresponding author: Position and Special field of the first author
2
Position and Special field of the second author

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market place; random costs assumption is valid under the condition that market behavior can be fully
characterized in the context of probability measures.

In this paper, first in section2, we describe some basic definitions of fuzzy theory like fuzzy numbers, fuzzy
random variables … .In section 3, we discuss the renewal reward process when the reward and inter arrival time
cannot be known exactly. We regard to the prices as fuzzy numbers and the life time of a switch as a fuzzy
random variable. Then we prove a theorem about the long-run average fuzzy reward and fuzzy inter arrival time.
In section 4, we introduce a T-age replacement policy and we propose to find T age such that coast to be
minimum. Finally in section 5 an example is considered.

2. Fuzzy numbers and fuzzy variables

Some basic definitions of fuzzy numbers and fuzzy random variables are stated here.

If  is some set, then a fuzzy subset A of  is defined by its membership function, written by  A (x) , which
produces values in [0,1] for all x in  . So,  A (x) is a function mapping  into [0,1] . When  A (x) is always
equal to one or zero we obtain a crisp (nonfuzzy) subset of  .

A general definition of fuzzy number may be found in ([4],[5]), however our fuzzy numbers will be almost,
fuzzy numbers. A triangular fuzzy number N is defined by three numbers a < b < c where the base of the
triangle is the interval [a, c] and its vertex is at x = b. Triangular fuzzy numbers will be written as N = (a/b/c).

Alpha-cuts are slices through a fuzzy set producing regular (non-fuzzy) sets. If A is a fuzzy subset of some set
 , then an  -cut of A , written A[ ] is defined as

A[ ]  {x   |  A ( x)  }, (1)

for all  ,0    1 .

A is called convex fuzzy set if  A [x  (1   ) y]  min{A ( x), A ( y)} for  [0,1].
Let A be a fuzzy set with membership function  A (x) and A[ ]  {x   |  A ( x)  }, Then
 A ( x)  sup 1A ( x), (2)
[0,1]
where
1, x  A
1A ( x)   (3)
o, o.w.

Fuzzy variable: Let  be a nonempty set, and P() shows the power set of  . In order to present the axiomatic
definition of possibility, Nahmias [6] and Liu [7] gave the following four axioms:
Axiom 1. Pos{}  1.
Axiom 2. Pos{}  0.
Axiom3. Pos{ i Ai }  supi Pos{Ai } for any collection Ai in P() .
Axiom 4. Let  i be nonempty sets on which Posi {.}satisfies the first three axioms, i=1,2,...,n, respectively, and
  1  2  ... n . Then
Pos{ A}  sup Pos1{1}  Pos 2 { 2 }  ... Posn { n } (4)
(1 ,...,n )A

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for each A  P().

Definition 1. (Liu and Liu [8]). Let  be a nonempty set, and P() the power set of  . Then the set function
Pos is called a possibility measure, if it satisfies the first three axioms, and (, P(), Pos) is called a possibility
space.

Definition 2. Let (, P(), Pos) be a possibility space, and A presents a set in P() . Then the necessity measure
of A is defined by:
Nec{A}  1  Pos{Ac }. (5)
Definition 3. (Liu and Liu [8]). Let (, P(), Pos) be a possibility space, and A be a set in P() . Then the
credibility measure of A is defined by:
1
Cr{ A}  ( Pos{ A}  Nec{ A}). (6)
2
Definition 4. A fuzzy variable  is defined as a function from the possibility space (, P(), Pos) to the set of
real numbers, and its membership function is derived by:
 (r )  Pos{   |  ( )  r}. (7)
Definition 5. (Liu and Liu [8]). Let  be a fuzzy variable on the possibility space (, P(), Pos) . Then the
expected value E[ ] is defined by:
 0


0

E[ ]  Cr{  r}dr  Cr{  r}dr ,

(8)

provided that at least one of the two integrals is finite. In particular, if the fuzzy variable  is positive (i.e.
Pos{  0}  0 ), then
0
E[ ]   Cr{  r}dr.

(9)

3. Fuzzy renewal reward processes

~ ~
Consider a renewal process N (t), t > 0 having inter arrival times X n for n  1 , and suppose that X n is a random
~
fuzzy variable and each time a renewal occurs we receive a reward. We denote by Rn , the fuzzy reward earned
~ ~
at the time of the n-th renewal, where Rn is a fuzzy random variable. We shall assume that the Rn for n  1 are
independent and identically distributed. If we let
~ ~
R (t )   nN(1t ) Rn , (10)
~ ~ ~
then R (t ) represents the total fuzzy reward earned by time t. Then, ( Rn )L and ( Rn )U are random variables for all
 [9]. Thus we have

~ ~
E[ RL ]  E[( Rn )L ], (11)
~ ~
E[ RU ]  E[( Rn )U
 ], (12)
~ ~
E[ X ]  E[ X n ]. (13)

~ ~ ~ ~
Theorem 3.1 Let AL  E[ RL ] / E[ X ] and AU  E[ RU ] / E[ X ] . Let A be a fuzzy number with membership
function

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 A (r )  sup I [ AL , AU ] (r ). (14)
 
0 1
~ ~ ~ ~ ~
Suppose that E[ RL ]   , E[ RU ]   and E[X ]   for all  . Then R (t ) / t  A with probability one level-wise .
If AL and AL are left continuous with respect to  .
~
Proof. We can rewrite R (t ) / t to following form

N (t ) N (t )

 ( R )  ( R )
~ L ~ L
n n
~ n 1 N (t )
( R (t ) / t )L   [ n1 ][ ].
t N (t ) t
By the strong law of large numbers, we obtain that
N (t )

 ( R )
~ L
n
~
[ n 1 ]  E[ RL ]
N (t )
with probability one as t   .
Similarly, we have

N (t )

 ( R )
~ U
n
~
[ n1 ]  E[ RU ]
N (t )
with probability one as t   .
Applying the fact that

N (t ) 1
 ~
t E[ X ]

([10]) with probability one as t   we get the desired result.■

We say that a cycle is complete every time if a renewal occurs, then Theorem 3.1 presents the long-run average
~
fuzzy reward when only the cycle is considered since E[X ] is the expected length of the cycle.

4. T-age replacement policy

The renewal reward theorem is the key tool for obtaining maintenance or replacement strategies which have
small long-run cost associated with them. Consider a system of one component which fails over time. The
maintenance policy of interest is “Replace the item at time T or at failure, whichever comes first”. Such policy
might be appropriate, for instance, in the following situation. Consider a production process which gets initiated
every day by a switch. The switch fails and needs to be replaced periodically. There are two costs associated
with this strategy: c1 which is the cost of buying a new switch and an additional cost of c 2 if the switch fails.
Suppose that c1 and c 2 cannot be known precisely. Then it will be more reasonable to assume c1 and c 2 as
closed fuzzy numbers c1 and c 2 , respectively.

~
We say that a cycle is complete every time if a new item is purchased. Let X be the lifetime of the item and be
a fuzzy random variable. Then the fuzzy cost incurred during a cycle will be given by

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~
~  c, X T
R 1 ~ (15)
c1  c2 , X T

Then, we have
 c L , ~
~ X T
RL   L 1 L ~ (16)
c1  c 2 , X T

 c U , ~
~ X T
RU   U 1 U ~ (17)
c1  c 2 , X  T
Then
~ ~ ~
E[ R ]  c1Cr{X  T }  (c1  c2 )Cr{X  T }
~ ~ ~
 c1 (Cr{X  T }  Cr{X  T })  c2Cr{X  T },
 c1  c2 FX~ (T ) (18)
Then
~
E[ RL ]  c1L  c2L FX~L (T ), (19)

~
E[ RU ]  c1U  c2U FXU
~ (T ),
 (20)

where FX~ (t ) is the cumulative distribution function of the fuzzy random variable of X. If c1 and c 2 are canonical
~ ~
fuzzy numbers then E[ RL ] and E[ RU ] are continuous with respect to  . The length of the cycle is
 X~ , ~
X T
 ~ (21)
 T , X T
The expected length of the cycle is
T

 Cr{X  r}dr  T (1  F (T )),

~
(22)
0

Therefore by theorem 3.1, we have

R (t ) L c1  c2 FX~ (T )
L L L
( )  ~ as t  , (23)
t E[ X L ]
and
R (t ) U c1  c2 FX~ (T )
U U U
( )  ~ as t  , (24)
t E[ X U ]

5. Numerical results
Suppose that experience shows that the switch will fail approximately between the 0th and 10th day. We then
assume that the lifetime of the switch is fuzzy uniform random variable on ( 0,10) . Then
a  0  (0.1,0,0.1)
and

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
b  10  (9.5,10,10.5).
Suppose that c1 and c2 are triangular fuzzy numbers with
 
c1  2000  (1800,2000,2200)
and

c2  300  (280,300,320).

Now we have
aL  0  0.1 , aU  0  0.1 , bL  10  0.5 ,
bU  10  0.5 and c1L  2000  200 ,
c1U  2000  200 , c2L  300  50 ,
c2U  300  50 ,
Then we have
~ 10  0.6
E[ X L ]   T (1  FL (T )),
2
~ 10  0.6
E[ X U ]   T (1  FU (T )).
2
So,
R (t ) L c1L  c2L FX~L (T )
( )  ,
t 10  0.6
 T (1  F (T ))
L
2

R (t ) U c1U  c2U FXU~ (T )

( )  .
t 10  0.6
 T (1  FU (T ))
2
Suppose that T = 7 years. Then

7 7
FX~ (7)  [ , ],
10.6  0.6 9.9  0.1

Thus,

7
1800  200  (280  20 )( )
AL  10.6  0.6 ,
9.4  0.6 7
 7(1  )
2 10.6  0.6

7
2200  200  (320  20 )( )
AU  9.9  0.1 .
10.6  0.6 7
 7(1  )
2 9.9  0.1

It is easy to see that AL and AU are continuous with respect to  . Then the membership function of A is

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Behrouz Fathi Vajargah, Sara Ghasemalipour/ TJMCS Vol. 4 No. 3 (2012) 402 - 410

 A (r )  sup 1[ AL , AU ] (r )  max 1[ AL , AU ] (r )
[0,1]   0 1  

For the finding the membership function we can use from the following method:

We should solve the nonlinear programming problem [11]

 A (r )  max 

Subject to

  1,
AL  r ,
AU  r ,
  0.

Since AL is increasing with respect to a and AU is decreasing with respect to  , we have:
(i) if A1L  r  A1U then  (r )  1 .
(ii) if r  A1L then  (r )  max{ : 0    1, a is the root of AL  r  0} .
(iii) if r  A1U then  (r )  max{ : 0    1, a is the root of AU  r  0} .
for this example, the result are

(i) A1L  A1U  311.268.

r  280.355
(ii) If r  311.268, then the membership of r is  (r )  .
30.911

330.105  r
(iii) If r  311.268, then the membership of r is  (r )  .
18.837

Table 1 and figure 1 shows the membership associated with the long-run average cost in this case.

Table 1. Membership function of the long-run average cost for T = 7

Long-run Membershi
average cost p function

282 0.0532
285 0.1503
290 0.3120
295 0.4737
300 0.6355

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Behrouz Fathi Vajargah, Sara Ghasemalipour/ TJMCS Vol. 4 No. 3 (2012) 402 - 410

311.268 1.0000
315 0.8019
320 0.5364
325 0.2710
330 0.0056

Figure 1. Membership function for the long-run cost when T = 7

6. Conclusion
In this paper, we discussed a renewal reward theorem which shows the asymptotic behavior of the
fuzzy lifetime distribution and the fuzzy expected cost per unit time and a numerical procedure to
calculate the membership associated with the long-run average fuzzy cost are provided. The presented
methodology works for any fuzzy lifetime distribution and closed fuzzy numbers failure costs. In our
future work, it could extend this approach to systems with more than one unit and policies like m-
failure or block replacement.

References
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deteriorating singleunit systems'', Naval Research Logistics 36 (1989) 419-446.
[4] J.J. Buckley, ''Fuzzy Probability and Statistics'', Studies in Fuzziness and Soft Computing, Volume
196 (2006) 8-12.

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[5] P. Chang, ''Fuzzy strategic replacement analysis'', European Journal of Operational Research
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