Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 586 https://internationalpubls.com A Fuzzy Optimal Base Stock System For Tolerant Clients a Jagatheesan. R *, b Ramesh. R, c Seenivasan. M, d Rajasekar. S. P a Department of Statistics, Salem Sowdeswari College, Salem, Tamil Nadu, India. *Corresponding author Email: drrjstat@gmail.com b Department of Mathematics, Arignar Anna Govt. Arts College, Musiri, Tamil Nadu, India. Email: rameshsanju123@gmail.com c Mathematics Wing - CDOE, Annamalai University, Annamalai Nagar, Tamil Nadu, India. Email: emseeni@yahoo.com d Department of Mathematics, Government Arts College for Women, Nilakkottai, Dindigul, Tamil Nadu, India. Email: sekaraja.sp@gmail.com Article History: Received: 12-04-2024 Revised: 09-06-2024 Accepted: 22-06-2024 Abstract: Base stock is an amount of stock that a company requires to maintain in order to handle an unexpected huge demand. To lerant clients are the clients who are tolerant to the company’s delayed supply, who come back to the same company even if their demands are not met within their expected time. This implies that the clients are solely relied on that company. Marketing manager requires to maintain on hand to satisfy such tolerant clients’ demands with some delay, no longer than expected by that tolerant client.This paper diagnosis the performance measures of optimal base stock system for tolerant clients in fuzzy environment. The fuzzy numbers can be modified to crisp number with the help of fuzzy ranking methods. Here the used fuzzy ranking method is prominent for de-fuzzification. The famous Triangular fuzzy number and Trapezoidal fuzzy number methods are played a major role for de-fuzzification. At last, the optimization of base stock is verified with numerical examples in fuzzy environment. Keywords: Base stock, Tolerant clients, Fuzzy environment, Fuzzy ranking, De- fuzzification. 1. Introduction In the theory of sock management, the well-known base stock system for tolerant clients is an important model and these models have been considered by many researchers. Base Stock system for tolerantclients in the theory of sock management is an absorbing method of ordering techniques. Initially the stocks starting with ‘B’ number of items. Whenever a client order for ‘r’ units is received, astock replenishment order for ‘r’ units is placed instantly. Replenishment orders are full filled after the lead time ‘L’. The client’s demand is met with viable from the supply on hand. If the total unfilled client’s demand surplus the supply on hand, then assume that the client will not cancel the orders but the clients wait till their requirement is fulfilled. The sum of stock on hand and order placed is constant in time and equal to ‘B’ called as Base Stock. The detailed study of the base stock system for tolerant clients has been considered by Gaver[8]. The very basic model had been discussed by Hanssman [9]. Ramanarayanan et al., [18] have discussed the model in which the lead time was assumed to be a random variable. Suresh Kumar [26] have proposed the model by using the change of distribution property. Baimei Yang et al [1] derived a mailto:drrjstat@gmail.com mailto:rameshsanju123@gmail.com mailto:emseeni@yahoo.com mailto:sekaraja.sp@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 587 https://internationalpubls.com single period inventory model with two suppliers. Ramathilagam et al., [19] proposed an optimal reserve inventory model with the assumption that it undergoes parametric change. Chung – Ho Chen [6] have developed the optimum production run length using SCBZ property. ShirajulIslam Ukil et al., [25] have proposed a production inventory model with the assumptions that the production and rate of demand are in linear trend. The concept of SCBZ property was basically discussed by RajaRao and Talwalker [17]. Henry et al [10, 11, 12, 13] have discussed the optimum base stock model with the assumption that the lead time random variable has the SCBZ property and assuming that the truncation point itself a random variable which follows exponential distribution. Kaufmann [14] explained in detail the theory of fuzzy subsets. Campos L et al., [2], Cheng CH [3], Chu TC [4] and Chen et al., [5] are developed their models by applying the ranking fuzzy numbers. The theory of fuzzy sets and fuzzy logic was developed by Klir et al., [15] and Zadeh [33, 34]. Modarres et al., [16] have explained in detail in the performance ratio by using the concept of fuzzy ranking numbers. Ramesh et al [20, 21] have applied the very famous wingspans fuzzy ranking method for developing M/M/1/N fuzzy queuing model. Syamala et al., [27, 28, 29, 30] discussed about some fuzzy concepts. Wang Y et al., [31], Deng Y et al., [7]and Yager R R[32] are introduced the revised fuzzy ranking methods for their articles. Sachithanantham et al., [22,23, 24] have discussed the modified version of the Base Stock model in which the lead time was assumed to be a random variable and which satisfies the so called SCBZ property. In the sense that after the truncation point the lead time takes parametric change. The change point known as the truncation point and is also assumed to be a random variable and based on these assumptions, optimal Base Stock has been obtained. Also it is assumed that the lead time random variable follows exponential distribution which satisfies the SCBZ property and the change point is itself a random variable which has the mixed exponential distribution. Under this assumption, the optimal Base Stock is obtained. This article considers the above said model and verified their performance in fuzzy environment. The fuzzy numbers can be modified to crisp number with the help of Wingspans fuzzy ranking method. Here the well-known Triangular fuzzy numbers and Trapezoidal fuzzy numbers are played for de-fuzzification. Finally, the optimization of base stock is verified with a numerical example and to validate of our proposed method. 2. Preliminaries Definition 1: Base Stock Base stock is an amount of stock that marketing requires to maintain on hand to satisfy clients’ demands with some delay, no longer than expected by clients. Definition 2: Tolerant Clients Tolerant clients are clients who are tolerant to the company’s delayed supply. They come back to the same company even if their demands are not met within their expected time, which implies the clients are solely relied on that company. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 588 https://internationalpubls.com Definition 3: Fuzzy Number A fuzzy number is a generalization of a regular real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. Definition 4: Triangular Fuzzy Number A triangular fuzzy number A ~ (x1, x2,x3; 1) is dictated by a membership function           − − =  − − = otherwise aza aa az az aza aa az z A ,0 , , 1 , )( 32 32 3 2 21 12 1 ~ Definition 5: Trapezoidal Fuzzy Number A trapezoidal fuzzy number A ~ ( 1a , 2a , 3a , 4a ; 1) is dictated by a membership function           − −   − − = otherwise aza aa az aza aza aa az z A ,0 , , 1 , )( 43 43 4 32 21 12 1 ~ 3. Model Description The model which is assumed that the lead time random variable follows an exponential distribution, which satisfies the so called SCBZ property and it is also assumed that the truncation point itself a random variable which has the mixed exponential distribution with parameter τ and δ. Based on the above stated assumptions the optimal base stock model considered by Sachithanantham et al., and the derived model is 𝑑 ℎ+𝑑 − { 𝜆𝛽 [1−𝑒 −𝜇𝐵( 𝜃+τ 𝜆+𝜃+τ ) ] (𝜃+τ)(𝜆+𝜃+τ) } {𝜃 − 𝜃∗τ (𝜃+τ−𝜃∗) } = { 𝜆(1−𝛽)[1−𝑒 −𝜇𝐵( 𝜃+δ 𝜆+𝜃+δ ) ] (𝜃+δ)(𝜆+𝜃+δ) } {𝜃 − δ𝜃∗ (𝜃+δ−𝜃∗) } + { 𝜆[1−𝑒 −𝜇𝐵( 𝜃∗ 𝜆+𝜃∗) ] (𝜆+𝜃∗) } { τ𝛽 (𝜃+τ−𝜃∗) + δ(1−𝛽) (𝜃+δ−𝜃∗) }--------------------(1) Where h: Inventory holding cost / unit / time, d: Shortage cost / unit / time. B: The Base Stock level, λ: Parameter of inter arrival time distribution. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 589 https://internationalpubls.com µ: Parameter of demand distribution, τ, δ: Parameters of mixed exponential distribution. Here we have to verify the performance of optimum base stock model in fuzzy environment by using Wingspans ranking function method. 4. Wingspans Ranking Function Method Let a membership function A ~ of a fuzzy number A ~ which has the core point a0. Then WA =   − +− 0 0 )( 2 1 )( 2 1 ~~0 a A a A dxxdxxa  is called the Wingspans center of A ~ . This wingspans center based on the area between horizontal real axis and the curve of the membership function, also it is symmetric. The ranking function for triangular fuzzy number A ~ = [al, a0,ar] with respect to its wingspans center is R( A ~ ) = )( 4 1 2 1 0 rl aaa ++ . The ranking function for trapezoidal fuzzy number A ~ = [al,ab,ac,ar] with respect to its wingspans center is R( A ~ ) = )( 4 1 rcbl aaaa +++ We use this proposed fuzzy ordering technique, which de-fuzzifies the fuzzy numerals in the Optimum Base Stock model. We have used these fuzzy numerals as the fuzzy parameters. 5. Practical Examples Steel authority of India Ltd., Salem branch is one of the major sources from which various stainless steel vessel making companies get their supply. Consider the sales during marriage seasons or any festival seasons, in Steel authority of India Ltd where in the demand from their clients exceeds the stock level, there forth they could not satisfy their clients when they place excess orders.In such cases the marketing manager decides to maintain the optimum base stock level in-order to satisfy the tolerant clients. If the given situation is applied to a normal environment the above equation (1) is already derived and verified by numerical example. Here the same situation is applied in fuzzy environment and their performances are measured. For Triangular fuzzy number Under the triangular fuzzy numbering, the optimal base stock is measured by using the following illustration. Let us assumes the values of various parameters in equation (1) and its performance were analysed in Fuzzy environment. Illustration 1.1 Ford̃=[2,3,4],�̃�=[1,2,3],β̃=[0.2,0.5,0.8],�̃�=[0.5,1,1.5],δ̃=[0.5,1.5,2.5],θ̃ = [0.2,0.5,0.8], θ∗̃=[0.5,1,1.5],µ̃=[0.5,1.5,2.5], the optimal value of �̂�is obtained and the variations in �̂� for the changes in the value of h̃ are listed. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 590 https://internationalpubls.com Now the ranking index of h̃is R(h̃) = R(4,5,6) = 1 2 (5) + 1 4 (4 + 6)= 5, R(h̃) = R(9,10,11) = 10, R(h̃) = R(14,15,16) = 15, R(h̃) = R(19,20,21) = 20 By using the above de-fuzzified numbers for the stock holding cost and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. Table 1.1:The variations in �̂� for the changes in the value of h̃ Table 1.1 determines that if the de-fuzzified values of stock holding cost increased then the optimal base stock �̂� decreased. Fig.1.1:The variations in �̂� for the changes in the value of h̃ From the graph, it is observed that if the values of the shortage cost h̃ increases, then the optimum base stock is reduced. Illustration 1.2 For h̃=[4,5,6],�̃�=[1,2,3],β̃=[0.2,0.5,0.8], �̃�=[0.5,1,1.5],δ̃=[0.5,1.5,2.5], θ̃ = [0.2,0.5,0.8], θ∗̃=[0.5,1,1.5], µ̃=[0.5,1.5,2.5], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of d̃ are listed. Now the ranking index of d̃is R(d̃)= R(2.5,3.0,3.5) = 1 2 (3) + 1 4 (2.5 + 3.5)= 3, R(d̃) = R(4.5,5.0,5.5) = 5, R(d̃) = R(6.5,7.0,7.5) = 7, R(d̃) = R(8.5,9.0,9.5) = 9 By using the above de-fuzzified numbers for the stock shortage cost and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. �̃� �̃� µ̃ �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [4,5,6] [0.5,1.5,2.5] [0.5,1.5,2.5] [2,3,4] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 16207 [9,10,11] [0.5,1.5,2.5] [0.5,1.5,2.5] [2,3,4] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 8770 [14,15,16] [0.5,1.5,2.5] [0.5,1.5,2.5] [2,3,4] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 6040 [19,20,21] [0.5,1.5,2.5] [0.5,1.5,2.5] [2,3,4] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 4610 0 5000 10000 15000 20000 5 10 15 20 B ̂x 1 0 -4 holding cost h ̃ optimal Base stock 𝐵 ̂ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 591 https://internationalpubls.com Table 1.2: The variations in �̂� for the changes in the value of d̃ Table 1.2 determines that if the de-fuzzified values of stock shortage cost increased then the optimal base stock �̂�increased. Fig 1.2: The variations in �̂� for the changes in the value of d̃ From the graph, it is observed that if the values of the shortage cost d̃ increases, then the optimum base stock is increased. Illustration 1.3 For h̃=[4,5,6],�̃�=[2,3,4],β̃=[0.2,0.5,0.8], �̃�=[0.5,1,1.5],δ̃=[0.5,1.5,2.5], θ̃ = [0.2,0.5,0.8], θ∗̃=[0.5,1,1.5], µ̃=[0.5,1.5,2.5], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of �̃� are listed. Now the ranking index of �̃�is R(�̃�)= R(1.8,2.0,2.2) = 1 2 (2) + 1 4 (1.8 + 2.2)= 2.0, R(�̃�)= R(2.3,2.5,2.7) = 2.5, R(�̃�) = R(2.8,3.0,3.2) = 3.0, R(�̃�)= R(3.3,3.5,3.7) = 3.5 By using the above de-fuzzified numbers for the inter arrival time and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and are presented in the table below. �̃� �̃� µ̃ �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [1.8,2.0,2.2] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 16207 [2.3,2.5,2.7] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 17930 [2.8,3.0,3.2] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 19860 [3.3,3.5,3.7] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 21910 Table 1.3:The variations in �̂� for the changes in the value of �̃� Table.1.3 determines that if the de-fuzzified values of inter arrival time increased then the optimal base stock �̂� increased. �̃� �̃� µ̃ �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [2.5,3.0,3.5] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 16207 [4.5,5.0,5.5] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 25160 [6.5,7.0,7.5] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 33830 [8.5,9.0,9.5] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [1,2,3] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 42970 0 20000 40000 60000 3 5 7 9 B ̂x 1 0 -4 𝑑 ̃ 𝐵 ̂ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 592 https://internationalpubls.com Fig 1.3: The variations in �̂� for the changes in the value of d̃ From the graph, it is observed that if the values of the shortage cost d̃increases, then the optimum base stock is increased. Illustration 1.4 For h̃=[4,5,6],�̃�=[2,3,4],β̃=[0.2,0.5,0.8], �̃�=[0.5,1,1.5],δ̃=[0.5,1.5,2.5], θ̃ = [0.2,0.5,0.8], θ∗̃=[0.5,1,1.5],�̃� =[1,2,3], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of µ̃ are listed.Now the ranking index of µ̃is R(µ̃)= R(1.3,1.5,1.7) = 1 2 (1.5) + 1 4 (1.3 + 1.7)= 1.5, R(µ̃) = R(1.8,2.0,2.2) = 2.0, R(µ̃) = R(2.3,2.5,2.7) = 2.5, R(µ̃) = R(2.8,3.0,3.2) = 3.0 By using the above de-fuzzified numbers for the demand and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. Table 1.4:The variations in �̂� for the changes in the value of µ̃ Table 1.4 determines that if the de-fuzzified values of demand increased then the optimal base stock �̂�decreased. Fig 1.4: The variations in �̂� for the changes in the value of d̃ From the graph, it is observed that if the values of the shortage cost d̃ increase, then the optimum base stock is increases. µ̃ �̃� �̃� �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [1.3,1.5,1.7] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 16207 [1.8,2.0,2.2] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 12160 [2.3,2.5,2.7] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 9720 [2.8,3.0,3.2] [0.5,1.5,2.5] [0.5,1.5,2.5] [4,5,6] [2,3,4] [0.2,0.5,0.8] [0.5,1,1.5] [0.2,0.5,0.8] [0.5,1,1.5] 8100 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 593 https://internationalpubls.com For Trapezoidal fuzzy number Let us assumes the values of various parameters in equation (1) and its performance were analyzed in Fuzzy environment. Illustration 2.1 For d̃=[1,2,3,4],�̃�=[1,2,3,4],β̃=[0.2,0.5,0.8,1.1], �̃�=[0.5,1.0,1.5,2.0],δ̃=[0.5,1.5,2.5,3.5],θ̃ = [0.2,0.5,0.8,1.1], θ∗̃=[0.5,1.0,1.5,2.0], µ̃=[0.5,1.0,1.5,2.0], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of h̃ are listed. Now the ranking index of h̃is R(h̃) = R(3,4,5,6) = 1 4 (3 + 4 + 5 + 6)= 4.5, R(h̃) = R(8,9,10,11) = 9.5, R(h̃) = R(13,14,15,16) = 14.5, R(h̃) = R(18,19,20,21) = 19.5 By using the above de-fuzzified numbers for the stock holding cost and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. Table 2.1: The variations in �̂� for the changes in the value of h̃ Table 2.1 determines that if the values of the holding cost h̃ increased then the optimum base stock is decreased. Fig.2.1:The variations in �̂� for the changes in the value of h̃ From the graph, it is observed that if the values of the holding cost h̃ increased then the optimum base stock is decreased. Illustration 2.2 Forh̃=[3,4,5,6],�̃�=[1,2,3,4],β̃=[0.2,0.5,0.8,1.1],�̃�=[0.5,1.0,1.5,2.0],δ̃=[0.5,1.5,2.5,3.5], �̃� �̃� µ̃ �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [3,4,5,6] [2,3,4,5] [0.5,1.0,1.5,2.0] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 22840 [8,9,10,11] [2,3,4,5] [0.5,1.0,1.5,2.0] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 8210 [13,14,15,16] [2,3,4,5] [0.5,1.0,1.5,2.0] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 5180 [18,19,20,21] [2,3,4,5] [0.5,1.0,1.5,2.0] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 3800 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 594 https://internationalpubls.com θ̃ = [0.2,0.5,0.8,1.1], θ∗̃=[0.5,1.0,1.5,2.0], µ̃=[0.5,1.0,1.5,2.0], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of d̃ are listed. Now the ranking index of d̃ R(d̃) = R(1,2,3,4) = 1 4 (1 + 2 + 3 + 4)= 2.5, R(d̃) = R(3,4,5,6) = 4.5, R(d̃) = R(5,6,7,8) = 6.5, R(d̃) = R(7,8,9,10) = 8.5 By using the above de-fuzzified numbers for the stock holding cost and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. Table 2.2: The variations in �̂� for the changes in the value of d̃ Fig 2.2:The variations in �̂� for the changes in the value of d̃ From the graph, it is observed that if the values of the shortage cost d̃increases, then the optimum base stock is increased. Illustration 2.3 For h̃=[3,4,5,6],d̃=[1,2,3,4],β̃=[0.2,0.5,0.8,1.1], �̃�=[0.5,1.0,1.5,2.0], δ̃=[0.5,1.5,2.5,3.5], θ̃ = [0.2,0.5,0.8,1.1], θ∗̃=[0.5,1.0,1.5,2.0], µ̃=[0.5,1.0,1.5,2.0], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of �̃� are listed. Now the ranking index of �̃� R(�̃�) = R(1,2,3,4) = 1 4 (1 + 2 + 3 + 4)= 2.5, R(�̃�) = R(2,3,4,5) = 3.5, R(�̃�) = R(3,4,5,6) = 4.5, R(�̃�) = R(4,5,6,7) = 5.5 By using the above de-fuzzified numbers for the stock holding cost and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. �̃� �̃� µ̃ �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [1,2,3,4] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 17200 [3,4,5,6] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 44090 [5,6,7,8] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 109990 [7,8,9,10] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 149980 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 595 https://internationalpubls.com Table 2.3: The variations in �̂� for the changes in the value of d̃ Fig 2.3:Variations in �̂� for the changes in the value of �̃� From the graph, it is observed that if the values of �̃� increased then the optimum base stock is reduced. Illustration 2.4 For d̃=[1,2,3,4],�̃�=[1,2,3,4],β̃=[0.2,0.5,0.8,1.1],�̃�=[0.5,1.0,1.5,2.0],δ̃=[0.5,1.5,2.5,3.5], θ̃ = [0.2,0.5,0.8,1.1], θ∗̃=[0.5,1.0,1.5,2.0], µ̃=[0.5,1.0,1.5,2.0], the optimal value of �̂� is obtained and the variations in �̂� for the changes in the value of µ̃ are listed.Now the ranking index of µ̃is R(µ̃) = R(0.5,1.0,1.5,2.0) = 1 4 (0.5 + 1.0 + 1.5 + 2.0)= 1.25, R(µ̃) = R(1.0,1.5,2.0,2.5) = 1.75, R(µ̃) = R(1.5,2.0,2.5,3.0) = 2.25, R(µ̃) = R(2.0,2.5,3.0,3.5) = 2.75 By using the above de-fuzzified numbers for the stock holding cost and similarly the de-fuzzified values of all other parameters, the optimal base stock �̂� is obtained by solving the equation (1) and presented in the table below. Table 2.4: The variations in �̂� for the changes in the value of µ̃ �̃� �̃� µ̃ �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [1,2,3,4] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 22840 [2,3,4,5] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 16860 [3,4,5,6] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 14860 [4,5,6,7] [2,3,4,5] [0.5,1.0,1.5,2.0] [3,4,5,6] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 13990 µ̃ �̃� �̃� �̃� �̃� �̃� �̃� �̃� 𝛉∗̃ �̂� x10-4 [0.5,1.0,1.5,2.0] [2,3,4,5] [3,4,5,6] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 22840 [1.0,1.5,2.0,2.5] [2,3,4,5] [3,4,5,6] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 16410 [1.5,2.0,2.5,3.0] [2,3,4,5] [3,4,5,6] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 12720 [2.0,2.5,3.0,3.5] [2,3,4,5] [3,4,5,6] [1,2,3,4] [1,2,3,4] [4,5,6,7] [3,4,5,6] [0.5,1.0,1.5,2.0] [3,4,5,6] 10390 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 596 https://internationalpubls.com Fig 2.4:The variations in �̂� for the changes in the value of µ̃ From the graph, it is observed that if the values of µ̃ increased then the optimum base stock is decreased. 6. Findings and Conclusions From the tables and graphs, it is observed that the performance of base stock is verified under fuzzy environmentsand it is found that, as the holding cost h̃ increases, the optimal Base Stock B ̂ decreases. As the shortage cost �̃� increases, the optimal Base Stock B ̂ increases.As�̃�, the parameter of inter arrival time distribution increases, the optimal Base Stock B ̂decreases.Asµ̃, the parameter of demand distribution increases, the optimal Base Stock B ̂ decreases.Hence theperformance of optimal base stock system for tolerant clients in fuzzy environment is verified by using our proposed ranking procedures. It is observed that the findings of both methods are reflecting the same as in Fuzzy Environment. Based on these results, the manager can hold of the supreme conclusion in the sense that the maintenance of optimal base stock under fuzzy environment. 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