Investigation of fuzzy inventory model of type (s,S) with Nakagami distributed demands

1 Introduction

The Nakagami distribution has been proposed for modeling the fading of radio signals (Nakagami [8]). Although there are many distributions for modeling the fading of radio signals such as Rician, Weibull, Lognormal, Rayleigh and etc., Nakagami distribution is one of the most popular among these distributions (Tarique and Hasan [17]). Among them, the Nakagami distribution has been given a special attention for its ease of manipulation and wide range of applicability. More importantly, the Nakagami distribution has been found to have a very good fit for the mobile radio channels (Yacoub et al. [22]). There are many interesting studies relating to Nakagami distribution in the communication area. The Nakagami distribution has also been applied in many fields. For example, Dumane and Shankar [4], Shankar et al. [15] and Tsui et al. [18] applied Nakagami distribution to the context of ultrasonic imaging to represent the statistics of the envelope of the backscattered echo from tissues in medical imaging studies. Sarkar et al. [12, 13] used the Nakagami distribution to generate geomorphological instantaneous unit hydrograph in the field of hydrology. Kim and Latchman [7] analyzed the statistical characteristics of “Moving Pictures Expert Group” (MPEG) frame data by using the Nakagami distribution. Carcole and Sato [3] use the Nakagami distribution, which better takes into account the interference of different kind of signals in coda waves due to multipath propagation effects with the presence of velocity perturbations. Nakahara and Carcole [10] have shown the usefulness of the Nakagami distribution for modeling high-frequency seismogramenvelopes.

The mathematical definition of Nakagami distribution is given as follows.

A random variable X that follows a Nakagami distribution has the density function f ( x ) = 2 Γ ( β ) ( β λ ) β x 2 β - 1 exp ( - β λ x 2 ) , x > 0 ,(1.1)

where shape parameter β ≥ 0.5 and spread parameter λ > 0. The shape parameter β is labeled as ‘m’ and referred to as the fading parameter. The spread parameter λ represents the average power of envelop signal. The expected value and variance of the Nakagami distribution are given as follows: E ( X ) = Γ ( β + 0.5 ) / Γ ( β ) ( λ / β ) 1 / 2 ; V ( X ) = λ ( 1 - 1 β ( Γ ( β + 0.5 ) / Γ ( β ) ) 2 ) .

Nakagami distribution has close relation with the some other important distributions. For example, if Y is Gamma distribution with the parameters (β,  λ/β), then X = Y ^1/2 has Nakagami distribution with parameter (β,  λ). It collapses to the Rayleigh distribution when β = 1, and to the half-normal distribution when β = 0.5. For this reason, the interval 0.5 < β < 1 is sometimes referred to as the “pre-Rayleigh” range for the shape parameter, while β > 1 defines the “post-Rayleigh” range (Schwartz et al. [14]). Nakagami distribution has also a relation with Chi-Square distribution. Schwartz et al. [14] investigated maximum likelihood estimator of the shape parameter of the Nakagami distribution and they obtained some statistical properties of this estimator. Yacoub et al. [22] also studies the order statistics of the Nakagami distribution. We used Nakagami distribution for an inventory problem. Let us consider the inventory model of type (s, S) before stating the problem mathematically.

The model.

Let X (t) be the stock level in a depot at time t. As an assumption, the stock level X (t) at time t = 0 is S. The demands that come at random times are met with the amount S which is the starting stock level for first period. Stock level decreases according to demand quantities(- η _n ). These decreases happen at random times T n = ∑ i = 1 n ξ i and occur until random τ ₁ which means the stock level falls below to level s which is the control level. Here η _n , n = 1,2, …  ara random variables that express the demand quantity. Also ξ _n , n = 1,2, …  are random variables that express the inter-arrival times between consecutive demands. When the stock level fall below to s, instantaneously, the stock level is brought to level S. In other words, when the stock in the depot falls below to control levels, we refill the depot to the maximum stock level S. At the time, the first period of the process has been completed. Next demand quantities fall from maximum stock level S until stock level decreases below to stock control level s at time τ ₂ and this time stock level is again immediately brought to maximum stock level S. Then the second period has been completed. Afterward the process continues in similar way.

In this study, a stochastic process (X (t)) which describes an inventory model of type (s, S) is investigated where 0< s < S < ∞. There are many interesting studies on this topic in the literature (see, for example, Artelejo et al. [2], Nasirova et al. [11] and so on). Recently some studies on stochastic processes are made by using fuzzy logic (Zhao et al. [23], Li [8], Hong [5], Wang and Watada [19], Wang et al. [20], Wang [21], Khaniyev et al. [6]). For this reason, we construct the process X (t) under assumption that demands are random variables having Nakagami distribution with fuzzy spread parameter.

The rest of this study is organized as follows In Section 2 of this study, definition and construction of the process is given in detail. In Section 3, the α-cut of the ergodic distribution function of the process is obtained based on the monotonicity of the renewal function by using Zadeh’s extension principle. In Section 4, numerical study is given. Conclusions are summarized in Section 5.

To construct the process X(t), two independent sequences of positive valued random variables {ξ _n } and {η _n } ,  n ≥ 1 are defined on some probability space ( Ω , F , P ). Moreover, the variables in each sequence are independent and identically distributed. The random variables {η _n } ,  n ≥ 1 are interpreted as the amount of demands; {ξ _n } ,  n ≥ 1, are interpreted as the inter-arrival times between demands. In this study, we assume that the random variables η _n ’s have a Nakagami distribution with parameters (β, λ). Using the initial sequences {ξ _n } and {η _n }, one defines the renewal sequences {T _n }and {Y _n } as follows: T n = ∑ i = 1 n ξ i , Y n = ∑ i = 1 n η i , T 0 = 0 , n ≥ 1 , and construct the sequence of integer-valued random variables {N _n } as follows: N 0 = 0 ; N 1 = min { k ≥ 1 : S - Y k < s } ; n ≥ 1 ; N n + 1 = min { k ≥ N n + 1 : S - ( Y k - Y N n < s } .

Moreover, we put ν (t) = max{ k ≥ 0 : T _k  ≤ t }; τ n = T N n = ∑ i = 1 N n ξ i , n ≥ 1 ; τ 0 = 0 .

Now using the random variables defined above we can construct the desired stochastic process X (t) as follows: X ( t ) = S - ( Y ν ( t ) - Y N n ) , if τ n ≤ t < τ n + 1 , n ≥ 0 .

The random variables τ _n ,  n ≥ 0, are interpreted as the passage times to the control level s >  0 by the process X (t).

A sample trajectory of the process X(t) is given at Fig. 1.

Our purpose in this study is to calculate the membership function of the ergodic distribution of the process X(t) when parameter λ of the Nakagami distribution is a fuzzy number.

3 Main Results

Assume that Q (x) denotes the ergodic distribution of the process X (t), and U _η  (x) is the renewal function generated by the sequence η _n , n ≥ 1. In Nasirova et al. [11], the ergodic distribution of the process X (t) is found for the crisp case. We state this result as the following theorem without proof.

Proposition 3.1. (Nasirova et al. [11]) Let the initial sequences of the random variables {ξ _n } and {η _n }, additionally satisfy the following conditions:

0< E (ξ ₁) < ∞,

Then the process X (t) is ergodic and its ergodic distribution has the following explicit form: Q ( x ) = 1 - U η ( S - x ) U η ( S - s ) , s ≤ x ≤ S .(3.1)

When η _n ,  n = 1, 2, … have the Nakagami distribution with parameters (β, λ), instead of the renewal function of the U _η  (x), we use the notation U _β,λ (x).

Therefore, we can state the following proposition on the renewal function generating by Nakagami distribution.

Proposition 3.2. Assume that U_β,λ (x) andU_β,1 (x) are the renewal functions of the Nakagami distributions with parameters (β, λ) and (β, 1), respectively. Then for each ≥0, the following equality is holds: U β , λ ( t ) = U β , 1 ( t / λ ) .(3.2)

Proof. The random variables η _n , n = 1, 2, …, have Nakagami distribution with parameters (β, λ). Denote their distribution function by G _β,λ (t), then G β , λ ( x ) = γ ( β , β x 2 / λ ) Γ ( β ) where γ (β, t) and Γ (β) are incomplete and complete gamma functions, respectively (Abramowitz and Stegun [1]). From the definition of G _β,λ (x), the following equality can be obtained: G β , λ ( x ) = γ ( β , β x 2 / λ ) Γ ( β ) = γ ( β , β ( x / λ ) 2 ) Γ ( β ) = G β , 1 ( x / λ )

On the other hand, the renewal function U _η  (x) generated by the random variables η _n has the following form: U η ( t ) = ∑ n = 0 ∞ F * n ( t ) = ∑ n = 0 ∞ G β , λ * n ( t ) ≡ U β , λ ( t )

Since G β , λ ( x ) = G β , 1 ( x / λ ), then U β , λ ( t ) = ∑ n = 0 ∞ G β , 1 * n ( t / λ ) = U β , 1 ( t / λ ) .

Summarizing, the renewal function (U _β,λ (t)) of the Nakagami distribution with parameters (β, λ) is expressed by the renewal function of Nakagami distribution with parameters (β, 1), as follows: U β , λ ( t ) = U β , 1 ( t / λ ) .

This completes the proof of Proposition 3.2. ■

Remark. As seen from Proposition 3.2, U _β,λ (x) is monotone decreasing function with respect to parameter λ. For this reason, application of Zadeh’s Extension Principle is advisable in this work. As an example, plot of U _β,λ (t) is given at Fig. 2 for various values of λ when β = 05.

Now, we define the membership function of the fuzzy parameter λ ˜ as follows: μ λ ˜ ( x ) = { μ λ L ( x ) , μ λ R ( x ) , 0 , λ 1 ≤ x ≤ λ 2 λ 2 ≤ x ≤ λ 3 otherwise

Here, 0 < λ ₁ < λ ₂ < λ ₃ < ∞ ,  μ _λL  (x) is increasing function in the interval of [λ ₁, λ ₂] and μ _λR  (x) is decreasing function in the interval of [λ ₂,  λ ₃]. Moreover, μ _λL  (λ ₁) = μ _λR  (λ ₃) =0 and μ _λL  (λ ₂) = μ _λR  (λ ₂) =1. A sample plot of a membership function of μ λ ˜ ( x )is given at the Fig. 3.

Since U β , 1 ( t / λ )is decreasing function respect to λ, the membership function of the U β , 1 ( t / λ ˜ )can be presented as in the following proposition with the use of the extension principle of Zadeh.

Proposition 3.3. For each t >  0 the membership function of the U β , λ ˜ ( t ) can be expressed by means of the membership function of λ ˜ as follows: μ U β , λ ˜ ( t ) ( x ) = { μ λ R ( ( t / U β , 1 - 1 ( x ) ) 2 ) , x ∈ [ U β , 1 ( t / λ 3 ) , U β , 1 ( t / λ 2 ) ] μ λ L ( ( t / U β , 1 - 1 ( x ) ) 2 ) , x ∈ [ U β , 1 ( t / λ 2 ) , U β , 1 ( t / λ 1 ) ] 0 , otherwise

Here, U _β,1 (x) is the renewal function generated by Nakagami distribution with parameters (β, 1). □

Using the Proposition 3.3, we can calculate the α-cuts for U β , λ ˜ ( t ). Now, we give this result as the following proposition.

Proposition 3.4. The α –cuts of the U β , λ ˜ ( t ) can be presented as follows: [ U β , λ ˜ ( t ) ] α = [ U β , 1 ( t / μ λ R - 1 ( α ) ) ; U β , 1 ( t / μ λ L - 1 ( α ) ) ] .

Here [ A ˜ ] αshows the α-cuts of the fuzzy number A ˜.

  □

Now let us prepare the basis to calculate the membership function for ergodic distribution of the process X (t). According to Proposition 3.2, the following equalities can be written: U ( S - s ) = U β , λ ( S - s ) = U β , 1 ( ( S - s ) / λ ) U ( S - x ) = U β , λ ( S - x ) = U β , 1 ( ( S - x ) / λ ) , x ∈ [ s , S ] .

Taking into account the above notations, we can write: U ( S - x ) U ( S - s ) = U β , 1 ( ( S - x ) / λ ) U β , 1 ( ( S - s ) / λ )

According to the extension principle of Zadeh, we put the fuzzy number λ ˜ instead of the parameter λ in Proposition 3.1 and calculate the α-cuts of the following fraction: [ U β , 1 ( ( S - x ) / λ ) U β , 1 ( ( S - s ) / λ ) ] α = [ U β , 1 ( ( S - x ) / λ ) ] α [ U β , 1 ( ( S - s ) / λ ) ] α(3.3)

By applying the Proposition 3.4 to nominator and denominator of Equation (3.3):

[ U β , 1 ( ( S - x ) / λ ˜ ) ] α = [ U β , 1 ( ( S - x ) / μ λ R - 1 ( α ) ) ; U β , 1 ( ( S - x ) / μ λ L - 1 ( α ) ) ] α(3.4)

[ U β , 1 ( ( S - s ) / λ ˜ ) ] α = [ U β , 1 ( ( S - s ) / μ λ R - 1 ( α ) ) ; U β , 1 ( ( S - s ) / μ λ L - 1 ( α ) ) ] α(3.5)

According to the rule of the interval arithmetic, it is known that if a, b, c, d > 0 then [ a , b ] [ c , d ] = [ a d , b c ] .(3.6)

Using the Equations (3.6), (3.5) we can write the following equality:

[ U β , 1 ( ( S - x ) / λ ) U β , 1 ( ( S - s ) / λ ) ] α = [ U β , 1 ( ( S - x ) / μ λ R - 1 ( α ) ) U β , 1 ( ( S - s ) / μ λ L - 1 ( α ) ) ; U β , 1 ( ( S - x ) / μ λ L - 1 ( α ) ) U β , 1 ( ( S - s ) / μ λ R - 1 ( α ) ) ] α(3.7)

Since the Nakagami distribution satisfy the conditions of Proposition 3.1, we can use the Equation (3.1) for the ergodic distribution of X (t). Taking into account the Proposition 3.1, we can write α-cuts for the ergodic distribution Q ˜ ( x ) of the process X (t) with fuzzy spread parameter λ ˜ as follows: [ Q ˜ ( x ) ] α = [ Q ˜ ( x , α ) lower ; Q ˜ ( x , α ) upper ] ,(3.8) where, Q ˜ ( x , α ) lower = [ 1 - U β , 1 ( ( S - x ) / μ λ L - 1 ( α ) ) U β , 1 ( ( S - s ) / μ λ R - 1 ( α ) ) ] 0 1 ; Q ˜ ( x , α ) upper = [ 1 - U β , 1 ( ( S - x ) / μ λ R - 1 ( α ) ) U β , 1 ( ( S - s ) / μ λ L - 1 ( α ) ) ] 0 1 .(3.9)

Here [ x ] 0 1 ≡ min { 1 ; max { 0 ; x } } .

From Equation (3.8), we can derive the support set for the Q ˜ ( x ) when a ↘ 0as follows: support ( Q ˜ ( x ) ) = [ Q 1 ( x ) ; Q 3 ( x ) ] ,(3.10) where Q 1 ( x ) = [ 1 - U β , 1 ( ( S - x ) / λ 1 ) U β , 1 ( ( S - s ) / λ 3 ) ] 0 1 , Q 3 ( x ) = [ 1 - U β , 1 ( ( S - x ) / λ 3 ) U β , 1 ( ( S - s ) / λ 1 ) ] 0 1 .

Moreover, we denote the mode of the membership function of Q ˜ ( x ) by Q ₂ (x), i.e., Q 2 ( x ) = 1 - U β , 1 ( ( S - x ) / λ 2 ) U β , 1 ( ( S - s ) / λ 2 ) .(3.11)

Summarizing, we can state the following theorem on the ergodic distribution of the process X (t) when the parameter λ is a fuzzy number λ ˜.

Theorem 3.1. Let the conditions of Proposition 3.1 be satisfied and the random variable η ₁ have the Nakagami distribution with parameter ( β , λ ˜ ) , where λ ˜ is a fuzzy number with membership function μ λ ˜ ( x ) which is defined as in Equation (3.2).

Then for each s ≤ x ≤ S, the ergodic distribution function ( Q ˜ ( x ) )of the process X (t) with fuzzy parameter ( λ ˜ ) has the following α- cuts ( [ Q ˜ ( x ) ] α ): [ Q ˜ ( x ) ] α = [ Q ˜ ( x , α ) lower ; Q ˜ ( x , α ) upper ] , where Q ˜ ( x , α ) lower = [ 1 - U β , 1 ( ( S - x ) / μ λ L - 1 ( α ) ) U β , 1 ( ( S - s ) / μ λ R - 1 ( α ) ) ] 0 1 ; Q ˜ ( x , α ) upper = [ 1 - U β , 1 ( ( S - x ) / μ λ R - 1 ( α ) ) U β , 1 ( ( S - s ) / μ λ L - 1 ( α ) ) ] 0 1 .

Here, U _β,1 (x) is the renewal function generated by a Nakagami distribution with the parameter (β,1). ■

Let us now consider the following special case for obtaining membership function of Q ˜ ( x ), using theTheorem 3.1.

Special Case 3.1. Assume that λ ˜ is a triangular fuzzy number having the following membership function: μ λ ˜ ( x ) = { μ λ L ( x ) ≡ x - λ 1 λ 2 - λ 1 , x ∈ [ λ 1 , λ 2 ) μ λ R ( x ) ≡ λ 3 - x λ 3 - λ 2 , x ∈ [ λ 2 , λ 3 ) 0 , otherwise

Triangular fuzzy number λ ˜ can be shown as λ ₁ λ ₂ λ ₃.

Then, it can be written, μ λ L - 1 ( α ) = λ 1 + α ( λ 2 - λ 1 ) , μ λ R - 1 ( α ) = λ 3 - α ( λ 3 - λ 2 ) . α-cuts of the Q ˜ ( x )can be represented as follows: [ Q ˜ ( x ) ] α = [ Q ˜ ( x , α ) lower ; Q ˜ ( x , α ) upper ] , where Q ˜ ( x , α ) lower = [ 1 - U β , 1 ( ( S - x ) / λ 1 + α ( λ 2 - λ 1 ) ) U β , 1 ( ( S - s ) / λ 3 - α ( λ 3 - λ 2 ) ) ] 0 1 ; Q ˜ ( x , α ) upper = [ 1 - U β , 1 ( ( S - x ) / λ 3 - α ( λ 3 - λ 2 ) ) U β , 1 ( ( S - s ) / λ 1 + α ( λ 2 - λ 1 ) ) ] 0 1 .

Therefore, for each x ∈ (s, S) Q 1 ( x ) = [ 1 - U β , 1 ( ( S - x ) / λ 1 ) U β , 1 ( ( S - s ) / λ 3 ) ] 0 1 ; Q 2 ( x ) = 1 - U β , 1 ( ( S - x ) / λ 2 ) U β , 1 ( ( S - s ) / λ 2 ) ; Q 3 ( x ) = [ 1 - U β , 1 ( ( S - x ) / λ 3 ) U β , 1 ( ( S - s ) / λ 1 ) ] 0 1 .(3.12)

Thus, the membership function of the ergodic distribution Q ˜ ( x ) with the fuzzy parameter λ, can be represented as follows: μ Q ˜ ( x ) ( y ) = { y - Q 1 ( x ) Q 2 ( x ) - Q 1 ( x ) , y ∈ [ Q 1 ( x ) , Q 2 ( x ) ] Q 3 ( x ) - y Q 3 ( x ) - Q 2 ( x ) , y ∈ [ Q 2 ( x ) , Q 3 ( x ) ] 0 , otherwise

Q ₁ (x), Q ₂ (x) and Q ₃ (x) are defined as Equation (3.12).

Special Case 3.2. Assume that λ ˜ is a quadratic fuzzy number having the following membership function: μ λ ˜ ( x ) = { μ λ L ( x ) ≡ ( x - λ 1 λ 2 - λ 1 ) 2 , x ∈ [ λ 1 , λ 2 ) μ λ R ( x ) ≡ ( λ 3 - x λ 3 - λ 2 ) 2 , x ∈ [ λ 2 , λ 3 ) 0 , otherwise

Then, it can be written, μ λ L - 1 ( α ) = λ 1 + α ( λ 2 - λ 1 ) , μ λ R - 1 ( α ) = λ 3 - α ( λ 3 - λ 2 ) , and α-cuts of the Q ˜ ( x ) can be represented as follows: [ Q ˜ ( x ) ] α = [ Q ˜ ( x , α ) lower ; Q ˜ ( x , α ) upper ] , where Q ˜ ( x , α ) lower = [ 1 - U β , 1 ( ( S - x ) / λ 1 + α ( λ 2 - λ 1 ) ) U β , 1 ( ( S - s ) / λ 3 - α ( λ 3 - λ 2 ) ) ] 0 1 ; Q ˜ ( x , α ) upper = [ 1 - U β , 1 ( ( S - x ) / λ 3 - α ( λ 3 - λ 2 ) ) U β , 1 ( ( S - s ) / λ 1 + α ( λ 2 - λ 1 ) ) ] 0 1 .

Therefore, for each x ∈ (s, S) Q 1 ( x ) = [ 1 - U β , 1 ( ( S - x ) / λ 1 ) U β , 1 ( ( S - s ) / λ 3 ) ] 0 1 ; Q 2 ( x ) = 1 - U β , 1 ( ( S - x ) / λ 2 ) U β , 1 ( ( S - s ) / λ 2 ) ; Q 3 ( x ) = [ 1 - U β , 1 ( ( S - x ) / λ 3 ) U β , 1 ( ( S - s ) / λ 1 ) ] 0 1 .

In this section, we first obtain the numerical values of renewal functions U λ ( x ) ( λ = 1 : 11 ¯ ), for each x value from A ≡ { x : x = v / 1000 , v = 1 : 5000 ¯ } by using following Volterra integral equation; U λ ( x ) = 1 + ∫ 0 x U λ ( x - y ) f ( y ) dy where f (y) is the Nakagami probability density function with shape parameter β = 0.5. To obtain the numerical values of U _λ  (x),we used the following equation in n-th steps; U ˆ λ ( n ) ( x ) = { 1 + ∑ i = 0 [ x / h ] h × U ˆ λ ( n - 1 ) ( x - i × h ) f ( i × h ) , n > 1 1 , n = 1 until the difference between | U ˆ λ ( n ) ( x ) - U ˆ λ ( n - 1 ) ( x ) | < 0.001for h = 0.001. Then for each U λ ( x ) ( λ = 1 : 11 ¯ ), we fit the data to the following equation; U ˜ λ ( x ) = a 1 x + a 2 + 1 - a 2 + c 1 x + c 2 x 2 1 + d 1 x + d 2 x 2 + d 3 x 3 where a 1 = Γ ( β ) / ( Γ ( β + 0.5 ) ( λ / β ) 1 / 2 ) ; a 2 = [ Γ ( β ) / ( Γ ( β + 0.5 ) ( λ / β ) 1 / 2 ) ] 2 λ 2 .

Thus, we obtain coefficients of this equation for every U λ ( x ) ( λ = 1 : 11 ¯ ) as given at Table 1. To measure the goodness of fit, we use the maximum relative error between the data U ˆ λ ( x ) and estimated function U ˜ λ ( x ). The maximum relative error and minimum accuracy percentage are given as follows;

δ λ = max x ∈ A | U ˆ λ ( x ) - U ˜ λ ( x ) | U ˆ λ ( x ) × 100 % ; AP = 100 - δ λ .

The coefficients and maximum relative errors and minimum accuracy percentage are given at Table 1.

As seen from Table 1, all of the relative errors of the models are less than 0.21% . In other words, all of the minimum accuracy percentages are more than 99.79% . This demonstrates that the proposed formulas show good approximation for the values λ = 1 : 11 ¯.

In the rest of this study, we give numerical results for (s, S) model with s = 2.7640 (2E (η ₁)), S = 15.2017 (11E (η ₁)), when η ˜ 1 has Nakagami distribution with shape parameter β = 0.5 and triangular fuzzy spread parameter λ ˜ = 2 / 3 / 4 at Table 2. We take different x value from x = 2.7640 to 15.2017. We obtain Q ₁ (x), Q ₂ (x) and Q ₃ (x) values from these x values using Theorem 3.1 and Special case 3.1.

The plot of these values is given at Fig. 4.

As seen from Fig. 4, fuzzy bounds at high values of x give close results to their crisp results. On the contrary, fuzzy bounds at low values of x do not give close results to their crisp results at high values of x.

5 Conclusion

Nakagami distribution is one of the most suitable distributions among the various distributions for nonlinear transformations of random variables which have Gamma distribution. For this reason, in this study, an inventory model of type (s, S) is investigated under the assumption that the amount of demands (η _n ) have Nakagami distribution. We first showed the monotonicity property of the renewal function U _β,λ (t)according to the spread parameter λ. By using this property and Extension Principle of Zadeh, we obtain the membership function of the renewal function. Then, we obtain the membership function of fuzzy ergodic distribution of the stochastic process which describes inventory model of type (s, S) under Nakagami distribution with fuzzy spread parameter. The results, which we have obtained, show that fuzziness of ergodic distribution at smallvalues of x is stronger than when x is large.