In the interval (0, N), the inventory level gradually decreases to meet demands. During the period [0, nd], the inventory depletes due to the demand. In addition, during the period [nd, n1], the inventory depletes due to the demand and deteriorating. By this process, during the interval [n1, N], the inventory level reaches zero level at time n1 and then shortage is allowed to occur. The probability density function of Gumbel distribution is \( f(t)=\frac{1}{\sigma }{e}^{{\left\{\frac{\left(t-\mu \right)}{\sigma}\right\}}^{\xi +1}}{e}^{-{e}^{{\left\{\frac{\left(t-\mu \right)}{\sigma}\right\}}^{\xi +1}}},t>0 \), where μ ∈ R is a location parameter, σ ∈ (0, ∞) is the scale parameter, and ξ ∈ (−∞, ∞) is the shape parameter. The shape parameter ξ governs the tail behavior of the distribution. The family defined by ξ → 0 corresponds to the Gumbel distribution. The probability density function of Gumbel distribution corresponds to minimum value \( {f}_{\mathrm{min}}(t)=\frac{1}{\sigma }{e}^{\frac{\left(t-\mu \right)}{\sigma }}\ \mathrm{Exp}\left[-\mathrm{Exp}\left[\frac{\left(t-\mu \right)}{\sigma}\right]\right],t>0 \). And the cumulative distribution function \( {F}_{\mathrm{min}}(t)=1-\mathrm{Exp}\left[-\mathrm{Exp}\left[\frac{\left(t-\mu \right)}{\sigma}\right]\right],t>0. \)
Then the deterioration rate function defined by \( {\theta}_{min}(t)=\frac{1}{\sigma }{e}^{\frac{\left(t-\mu \right)}{\sigma }},t>0. \)
The boundary conditions \( {q}_1(0)={Q}_m,{q}_2\left({n}_1\right)={q}_3\left({n}_1\right)=0,{q}_3(N)=\overline{x}-{Q}_m \).
The differential equations for the instantaneous inventory level qi (t), 0 < t < N are given by Eqs. (1), (2), (5), and (6).
$$ \frac{d\ {q}_1(t)}{dt}=-\overline{D}=-\frac{x}{n_d}\ \mathrm{for}\ 0\le t\le {n}_d $$
(1)
$$ \frac{d\ {q}_2(t)}{dt}+\theta (t)\ {q}_2(t)=-\overline{D}=-\frac{x}{n_1-{n}_d}\ \mathrm{for}\ {n}_d\le t\le {n}_1 $$
(2)
The solutions of the above differential equations after applying the boundary conditions are given by the following equations:
$$ {q}_1(t)={Q}_m-\overline{D}\ t={Q}_m-\frac{x\ t}{n_d}\ \mathrm{for}\ 0\le t\le {n}_d $$
(3)
$$ {q}_2(t)=\frac{x\ \sigma }{\left({n}_1-{n}_d\right)}\left\{\mathrm{Exp}\left[-\mathrm{Exp}\left[\frac{\left(t-\mu \right)}{\sigma}\right]\right]\left(\mathrm{Ei}\left[{\mathrm{e}}^{\frac{t-\mu }{\sigma }}\right]-\mathrm{Ei}\left[{\mathrm{e}}^{\frac{n_1-\mu }{\sigma }}\right]\right)\right\}\ for\ {n}_d\le t\le {n}_1 $$
(4)
where the integral Ei(x) for (x) ≥ 0 is defined by \( \mathrm{Ei}(x)=\underset{-x}{\overset{\infty }{\int }}\frac{e^{-t}}{t}\ dt \) [14]
$$ \frac{d\ {q}_3(t)}{dt}=-\frac{\overline{D}}{1+\epsilon \left(N-t\right)}=-\frac{x-{Q}_m}{N-{n}_1}\left(\frac{1}{1+\epsilon \left(N-t\right)}\right)\ {n}_1\le t\le N $$
(5)
$$ \frac{d\ {q}_4(t)}{dt}=-\overline{D}\left(1-\frac{1}{1+\epsilon \left(N-t\right)}\right)=-\frac{x-{Q}_m}{N-{n}_1}\ \left(1-\frac{1}{1+\epsilon \left(N-t\right)}\right)\ {n}_1\le t\le N $$
(6)
hence,
$$ {q}_3(t)=-\frac{x-{Q}_m}{\epsilon\ \left(N-{n}_1\right)}\ \left\{\ln \left[1+\epsilon \left(N-{n}_1\right)\right]-\ln \left[1+\epsilon \left(N-t\right)\right]\right\}\ {n}_1\le t\le N $$
(7)
$$ {q}_4(t)=-\frac{x-{Q}_m}{\epsilon\ \left(N-{n}_1\right)}\left\{\epsilon \left(t-{n}_1\right)+\ln \left[1+\epsilon \left(N-t\right)\right]-\ln \left[1+\epsilon \left(N-{n}_1\right)\right]\right\}\ {n}_1\le t\le N $$
(8)
The model for crisp environment
The expected annual total cost for the cycle is composed of expected order cost, expected purchase cost, expected varying deteriorating cost, expected varying salvage cost, expected backorder cost, expected lost sales cost, and expected varying holding cost
$$ E(PC)={C}_p\ \underset{x=0}{\overset{\infty }{\int }}{\int}_0^Nx\ f(x)\ dt\ dx={C}_p\ N\underset{x=0}{\overset{\infty }{\int }}x\ f(x)\ dx $$
(10)
$$ E(DC)={C}_d\ {N}^{\beta }\ \left(\ {Q}_m-\underset{x=0}{\overset{Q_m}{\int }}{\int}_{n_d}^{n_1}\ D(t)\ f(x)\ dt\ dx\right)\kern0.5em ={C}_d\ {N}^{\beta}\left({Q}_m-\frac{1}{\left({n}_1-{n}_d\right)}\underset{x=0}{\overset{Q_m}{\int }}\left({n}_1-{n}_d\right)\ x\ f(x)\ dx\right)={C}_d\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right) $$
(11)
$$ E(VC)={C}_v\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right)={C}_d\ \gamma\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right) $$
(12)
$$ E(BC)={C}_b\left(\underset{Q_m}{\overset{\infty }{\int }}\underset{n_1}{\overset{N}{\int }}\ \left(-{q}_3(t)\right)\ f(x)\ dtdx\kern0.5em \right)\kern0.5em ={C}_b\ \left[\underset{Q_m}{\overset{\infty }{\int }}\frac{x-{Q}_m}{\epsilon\ }\left(\begin{array}{c}1-\frac{1}{\epsilon\ \left(N-{n}_1\right)}\ln \left[1+\epsilon \left(N-{n}_1\right)\right]\end{array}\right)\ f(x)\ dx\right] $$
(13)
$$ E(LC)={C}_L\underset{x={Q}_m}{\overset{\infty }{\int }}\underset{n_1}{\overset{N}{\int }}\left(-{q}_4(t)\right)\ f(x)\ dt\ dx\kern0.5em ={C}_L\left\{\underset{Q_m}{\overset{\infty }{\int }}\frac{x-{Q}_m}{\epsilon}\left[\frac{\epsilon\ \left(N-{n}_1\right)}{2}-\left(1-\frac{1}{\epsilon\ \left(N-{n}_1\right)}\ln \left[1+\epsilon \left(N-{n}_1\right)\right]\right)\right]f(x)\ dx\right\} $$
(14)
$$ E(HC)={C}_h\ {N}^{-\beta}\left[{\int}_0^{Q_m}\left(\underset{0}{\overset{n_d}{\int }}{q}_1(t)\ dt+\underset{n_d}{\overset{n_1}{\int }}{q}_2(t)\ dt\ \right)f(x)\ dx+\underset{Q_m}{\overset{\infty }{\int }}\underset{n_1}{\overset{N}{\int }}\left(-{q}_4(t)\right)\ dt\ f(x)\ dx\right]\kern1em ={C}_h\ {N}^{-\beta}\left\{\underset{0}{\overset{Q_m}{\int }}\left[\left({Q}_m\ {n}_d-\frac{x\ {n}_d}{2}\right)+{\int}_{n_d}^{n_1}\left\{\left(\sum \limits_{i=0}^{\infty}\left(\frac{\mathrm{Exp}\left[-i\ \left(\frac{t-\mu }{\sigma}\right)\right]}{i!}\right)\right)\left[\left(\left(\frac{\mathrm{Exp}\left[\mathrm{Exp}\left[\frac{n_1-\mu }{\sigma}\right]\right]}{\left(\mathrm{Exp}\left[\frac{\mathrm{n}1-\mu }{\sigma}\right]\right)}\right)\sum \limits_{n=0}^{\infty}\left(n!\mathrm{Exp}\left[-n\left(\frac{n_1-\mu }{\sigma}\right)\right]\right)\right)-\left(\left(\frac{\mathrm{Exp}\left[\mathrm{Exp}\left[\frac{t-\mu }{\sigma}\right]\right]}{\Big(\mathrm{Exp}\left[\frac{t-\mu }{\sigma}\right]}\right)\sum \limits_{j=0}^{\infty}\left(j!\mathrm{Exp}\left[-j\ \left(\frac{t-\mu }{\sigma}\right)\right]\right)\right)\right]\right\} dt\right]f(x)\ dx+\underset{Q_m}{\overset{\infty }{\int }}\frac{x-{Q}_m}{\epsilon}\left[\frac{\epsilon\ \left(N-{n}_1\right)}{2}-\left(\begin{array}{c}1-\frac{1}{\epsilon\ \left(N-{n}_1\right)}\ln \left[1+\epsilon \left(N-{n}_1\right)\right]\end{array}\right)\right]f(x)\ dx\right\} $$
(15)
From Eqs. (9), (10), (11), (12), (13), (14), and (15), the expected annual total cost for the cycle is composed of E(TC) = E(OC) + E(PC) + E(DC (N)) + E(VC(N)) + E(BC) + E(LC) + E(HC(N))
$$ {\displaystyle \begin{array}{c}E\left( TC\left({Q}_m,{n}_d,{n}_1,N\right)\right)={C}_o+{C}_p\ N\underset{x=0}{\overset{\infty }{\int }}x\ f(x)\ dx\\ {}+\underset{Q_m}{\overset{\infty }{\int }}\frac{\left(x-{Q}_m\right)}{\epsilon }\ \left[\ \left({C}_b-{C}_h\ {N}^{-\beta }-{C}_L\right)\left(1-\frac{\ln \left[1+\epsilon \left(N-{n}_1\right)\right]}{\epsilon \left(N-{n}_1\right)}\right)+\left({C}_h\ {N}^{-\beta }+{C}_L\right)\frac{\left(N-{n}_1\right)}{\ \epsilon}\right]f(x)\ dx\\ {}+{C}_h\ {N}^{-\beta}\left\{\underset{0}{\overset{Q_m}{\int }}\left[\left({Q}_m\ {n}_d-\frac{x\ {n}_d}{2}\right)+\underset{n_d}{\overset{n_1}{\int }}\left(\left(\sum \limits_{i=0}^{\infty}\left(\frac{\mathrm{Exp}\left[-i\ \left(\frac{t-\mu }{\sigma}\right)\right]}{i!}\right)\right)\left(\left(\frac{\mathrm{Exp}\left[\mathrm{Exp}\left[\frac{n_1-\mu }{\sigma}\right]\right]}{\left(\mathrm{Exp}\left[\frac{\mathrm{n}1-\mu }{\sigma}\right]\right)}\right)\times \sum \limits_{n=0}^{\infty}\left(n!\mathrm{Exp}\left[-n\left(\frac{n_1-\mu }{\sigma}\right)\right]\right)\right)\right.\right)\right.\\ {}\left.\left.\left.-\left(\left(\frac{\mathrm{Exp}\left[\mathrm{Exp}\left[\frac{t-\mu }{\sigma}\right]\right]}{\Big(\mathrm{Exp}\left[\frac{t-\mu }{\sigma}\right]}\right)\sum \limits_{j=0}^{\infty}\left(j!\mathrm{Exp}\left[-j\ \left(\frac{t-\mu }{\sigma}\right)\right]\right)\right)\right)\mathrm{d}t\right]f(x)\ dx\right\}\\ {}+{C}_d\ \left(1+\gamma \right)\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right)\end{array}} $$
(16)
This paper puts a constraint on varying deteriorating cost for if the deteriorating cost exceeds a certain limit, this tend to increase the expected annual total cost or lead to loss. The Karush–Kuhn–Tucker (KKT) conditions [15] are first-order necessary conditions for a solution in non-linear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. So, this method is suitable to solve this problem.
Consider a limitation on the expected varying deteriorating cost, i.e.:
$$ E(DC)={C}_d\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right)\le {k}_d $$
(17)
It may be written as Min E (TC(Qm, nd, n1, N.))
subject to inequality constraint E(DC(Qm, nd, n1, N)) ≤ kd.
To find the optimal values \( {Q}_m^{\ast },{n}_d^{\ast },{n}_1^{\ast } \), and N∗ which minimize E(TC) under the constraint (17), the Lagrange multipliers technique with the Kuhn-Tacker conditions [15] is used as follows:
$$ {\displaystyle \begin{array}{c}E\left(L\left({Q}_m,{n}_d,{n}_1,N\right)\right)={C}_o+{C}_p\ N\underset{x=0}{\overset{\infty }{\int }}x\ f(x)\ dx\\ {}+\underset{Q_m}{\overset{\infty }{\int }}\frac{\left(x-{Q}_m\right)}{\epsilon }\ \left[\left({C}_b-{C}_h\ {N}^{-\beta }-{C}_L\right)\left(1-\frac{\ln \left[1+\epsilon \left(N-{n}_1\right)\right]}{\epsilon \left(N-{n}_1\right)}\right)\right.\\ {}\left.+\left({C}_h\ {N}^{-\beta }+{C}_L\right)\frac{\epsilon\ \left(N-{n}_1\right)}{2}\right]f(x)\ dx\\ {}+{C}_h\ {N}^{-\beta}\left\{\underset{0}{\overset{Q_m}{\int }}\left[\left({Q}_m\ {n}_d-\frac{x\ {n}_d}{2}\right)\right.\right.\\ {}+\underset{\mathrm{n}\mathrm{d}}{\overset{\mathrm{n}1}{\int }}\left\{\left(\sum \limits_{i=0}^{\infty}\left(\frac{\mathrm{Exp}\left[-i\left(\frac{t-\mu }{\sigma}\right)\right]}{i!}\right)\right)\left[\left(\left(\frac{\mathrm{Exp}\left[\mathrm{Exp}\left[\frac{n_1-\mu }{\sigma}\right]\right]}{\left(\mathrm{Exp}\left[\frac{n_1-\mu }{\sigma}\right]\right)}\right)\right.\right.\right.\\ {}\left.\times \sum \limits_{n=0}^{\infty}\left(\begin{array}{c}n!\\ {} Exp\left[-{n}^{\ast}\left(\frac{n_1-\mu }{\sigma}\right)\right]\end{array}\right)\right)\\ {}\left.\left.\left.\left.-\left(\left(\frac{\mathrm{Exp}\left[\mathrm{Exp}\left[\frac{t-\mu }{\sigma}\right]\right]}{\Big(\mathrm{Exp}\left[\frac{t-\mu }{\sigma}\right]}\right)\sum \limits_{j=0}^{\infty}\left(j!\mathrm{Exp}\left[-j\ \left(\frac{t-\mu }{\sigma}\right)\right]\right)\right)\right]\right\}\mathrm{d}t\right]f(x)\ dx\right\}\\ {}+{C}_d\ \left(1+\gamma \right)\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right)+{\lambda}_d\left[\ {C}_d\ {N}^{\beta}\left({Q}_m-\underset{x=0}{\overset{Q_m}{\int }}x\ f(x)\ dx\right)-{K}_d\right]\end{array}} $$
(18)
The optimal values \( {Q}_m^{\ast },{n}_d^{\ast },{n}_1^{\ast } \), and N∗ can be calculated by setting the corresponding first partial derivatives of Eq. (18) equal to zero.
$$ \mathrm{Thus},\mathrm{let}\ g1\left({Q}_m,{n}_d,{n}_1,N\right)=\frac{\partial E\left(L\left({Q}_m,{n}_d,{n}_1,N\right)\right)}{\partial {Q}_m}\left|\begin{array}{c}\\ {}\\ {}{Q}_m={Q}_m^{\ast}\end{array}=0\right., $$
$$ g2\left({Q}_m,{n}_d,{n}_1,N\right)=\frac{\partial E\left(L\left({Q}_m,{n}_d,{n}_1,N\right)\right)}{\partial {n}_d}\left|\begin{array}{c}\\ {}\\ {}{n}_d={n}_d^{\ast}\end{array}=\right.0, $$
$$ g3\left({Q}_m,{n}_d,{n}_1,N\right)=\frac{\partial E\left(L\left({Q}_m,{n}_d,{n}_1,N\right)\right)}{\partial {n}_1}\left|\begin{array}{c}\\ {}\\ {}{n}_1={n}_1^{\ast}\end{array}=\right.0, $$
$$ g4\left({Q}_m,{n}_d,{n}_1,N\right)=\frac{\partial E\left(L\left({Q}_m,{n}_d,{n}_1,N\right)\right)}{\partial N}\left|\begin{array}{c}\\ {}\\ {}N={N}^{\ast}\end{array}=\right.0 $$
$$ and\ g5\left({Q}_m,{n}_d,{n}_1,N\right)=\frac{\partial E\left(L\left({Q}_m,{n}_d,{n}_1,N\right)\right)}{\partial {\lambda}_d}\left|\begin{array}{c}\\ {}\\ {}{\lambda}_d={\lambda}_d^{\ast}\end{array}=\right.0 $$
The goal is to solve the previous multivariable nonlinear system by using Newton’s method using Mathematica program. The following Algorithm is applied.
Step 1: Define G(y) and J(y): Let F be a function which maps ℝn to ℝn and The Jacobian matrix is a matrix of first-order partial derivatives without the equation of the constraint g5 to find the value of Kd as:
$$ G(y)=\left[\begin{array}{c}g1\left({Q}_m,{n}_d,{n}_1,N\right)\\ {}g2\left({Q}_m,{n}_d,{n}_1,N\right)\\ {}g3\left({Q}_m,{n}_d,{n}_1,N\right)\\ {}g4\left({Q}_m,{n}_d,{n}_1,N\right)\end{array}\right],J(y)=\left[\begin{array}{c}\begin{array}{cc}\frac{\partial g1}{\partial {Q}_m}& \frac{\partial g1}{\partial {n}_d}\\ {}\frac{\partial g2}{\partial {Q}_m}& \frac{\partial g2}{\partial {n}_d}\end{array}\ \begin{array}{cc}\frac{\partial g1}{\partial {n}_1}& \frac{\partial g1}{\partial N}\\ {}\frac{\partial g2}{\partial {n}_1}& \frac{\partial g2}{\partial N}\end{array}\\ {}\begin{array}{cc}\frac{\partial g3}{\partial {Q}_m}& \frac{\partial g3}{\partial {n}_d}\\ {}\frac{\partial g4}{\partial {Q}_m}& \frac{\partial g4}{\partial {n}_d}\end{array}\ \begin{array}{cc}\frac{\partial g3}{\partial {n}_1}& \frac{\partial g3}{\partial N}\\ {}\frac{\partial g4}{\partial {n}_1}& \frac{\partial g4}{\partial N}\end{array}\end{array}\right] $$
Step 2: Let y ∈ ℝn. Then y represents the vector \( \left[\begin{array}{c}\begin{array}{c}{Q}_m\\ {}{n}_1\\ {}{n}_d\\ {}N\end{array}\\ {}{\lambda}_d\end{array}\right] \)
Step3: Assume any initial value y0 for Qm, nd, n1, and N when λd = 0.
Step4: Calculate G(y0 ), J(y0) and then find the inverse matrix J−1(y0), for y0.
Step 5: Solve the system y1 = y0 − J−1(y0) G(y0).
Step 6: Use the results of y1 to find the next iteration y2 by using the same procedure.
Step 7: Keep repeating the process until finding the same results for two consecutive values of Qm, n1, nd, and N. Then from Eq. (17), it is possible to calculate Kd.
Step 8: Repeat steps 1, 2, 3, and 4 with changing λd and adding g5(Qm, nd, n1, N) to the system until obtaining the same results for two consecutive values of Qm, nd, n1, and N. Then these values are the optimal values of \( {Q}_m^{\ast },{n}_d^{\ast },{n}_1^{\ast },\mathrm{and}\ {N}^{\ast }. \)
Step 9: Thus, the optimal value of the annual expected total cost TC(Qm, nd, n1, N) can be easily calculated.
The model for fuzzy environment
The inventory cost coefficients, elasticity parameters, and other coefficients in the model are fuzzy in nature. Therefore, the decision variables and the objective function should be fuzzy as well. To solve this inventory model using Lagrange multiplier technique, it should be find the right and the left shape functions of the objective function and decision variables, by finding the upper bound and the lower bound of the objective function, i.e., \( {\overset{\sim }{L}}^L\left(\propto \right) \) and \( {\overset{\sim }{L}}^R\left(\propto \right) \). Recall that \( {\overset{\sim }{L}}^L\left(\propto \right) \) and \( {\overset{\sim }{L}}^R\left(\propto \right) \) represent the largest and the smallest values (the left and right ∝cuts) of the optimal objective function \( \overset{\sim }{L}\left(\propto \right). \) For example using approximated value of TFN of \( {\overset{\sim }{C}}_o \) which observe in Fig. 2.
Consider the model when all parameters are triangular fuzzy numbers (TFN) as given:
$$ {C}_p=\left({C}_p-{\omega}_1,{C}_p,{C}_p+{\omega}_2\right),\kern4.25em {C}_o=\left({C}_o-{\omega}_3,{C}_o,{C}_o+{\omega}_4\right), $$
$$ {C}_h=\left({C}_h-{\omega}_5,{C}_h,{C}_{hr}+{\omega}_6\right),\kern3.75em {C}_b=\left({C}_b-{\omega}_7,{C}_b,{C}_b+{\omega}_8\right), $$
$$ {C}_L=\left({C}_L-{\omega}_9,{C}_L,{C}_L+{\omega}_{10}\right)\kern1.75em and\kern1.75em {C}_d=\left({C}_d-{\omega}_{11},{C}_d,{C}_d+{\omega}_{12}\right). $$
where ωi, i = 1, 2, … … , 12 are arbitrary positive numbers under the following restrictions:
$$ 0\le {\omega}_1\le {C}_p,{\omega}_2\ge 0,\kern1.25em 0\le {\omega}_3\le {C}_o,{\omega}_4\ge 0,\kern1.25em 0\le {\omega}_5\le {C}_h,{\omega}_6\ge 0, $$
$$ 0\le {\omega}_7\le {C}_b,{\omega}_8\ge 0,\kern1em 0\le {\omega}_9\le {C}_L,{\omega}_{10}\ge 0\kern1.25em \mathrm{and}\kern1.25em 0\le {\omega}_{11}\le {C}_d,{\omega}_{12}\ge 0. $$
hence, the left and right limits of ∝cuts of Cp, Co, Ch, Cb, CL, and Cd are given by:
$$ {{\overset{\sim }{C}}_p}_L\left(\propto \right)={C}_p-\left(1-\propto \right){\omega}_1,\kern2.5em {{\overset{\sim }{C}}_p}_R\left(\propto \right)={C}_p+\left(1-\propto \right){\omega}_2, $$
$$ {{\overset{\sim }{C}}_o}_L\left(\propto \right)={C}_o-\left(1-\propto \right){\omega}_3,\kern2.5em {{\overset{\sim }{C}}_o}_R\left(\propto \right)={C}_o+\left(1-\propto \right){\omega}_4, $$
$$ {{\overset{\sim }{C}}_h}_L\left(\propto \right)={C}_h-\left(1-\propto \right){\omega}_5,\kern2.5em {{\overset{\sim }{C}}_h}_R\left(\propto \right)={C}_h+\left(1-\propto \right){\omega}_6, $$
$$ {{\overset{\sim }{C}}_b}_L\left(\propto \right)={C}_b-\left(1-\propto \right){\omega}_7,\kern2.25em {{\overset{\sim }{C}}_b}_R\left(\propto \right)={C}_b+\left(1-\propto \right){\omega}_8, $$
$$ {{\overset{\sim }{C}}_L}_L\left(\propto \right)={C}_L-\left(1-\propto \right){\omega}_9,\kern2.25em {{\overset{\sim }{C}}_L}_R\left(\propto \right)={C}_L+\left(1-\propto \right){\omega}_{10}, $$
and
$$ {{\overset{\sim }{C}}_d}_L\left(\propto \right)={C}_d-\left(1-\propto \right){\omega}_{11},\kern1.6em {{\overset{\sim }{C}}_d}_R\left(\propto \right)={C}_d+\left(1-\propto \right){\omega}_{12} $$
where
$$ {\overset{\sim }{C}}_p={C}_p+\frac{1}{4}\left({\omega}_2-{\omega}_1\right),\kern2.75em {\overset{\sim }{C}}_o={C}_o+\frac{1}{4}\left({\omega}_4-{\omega}_3\right), $$
$$ {\overset{\sim }{C}}_h={C}_h+\frac{1}{4}\left({\omega}_6-{\omega}_5\right),\kern2.7em {\overset{\sim }{C}}_b={C}_b+\frac{1}{4}\left({\omega}_8-{\omega}_7\right), $$
$$ {\overset{\sim }{C}}_L={C}_L+\frac{1}{4}\left({\omega}_{10}-{\omega}_9\right)\kern0.75em \mathrm{and}\kern1.25em {\overset{\sim }{C}}_d={C}_d+\frac{1}{4}\left({\omega}_{12}-{\omega}_{11}\right). $$
Likewise, the same steps which are taken in the crisp case can be applied here, except that the crisp costs of Cp, Co, Ch, Cb, CL, and Cd will be replaced by the fuzzy costs of \( {\overset{\sim }{C}}_p,{\overset{\sim }{C}}_o,{\overset{\sim }{C}}_h,{\overset{\sim }{C}}_b,{\overset{\sim }{C}}_L, and\ {\overset{\sim }{C}}_d \). Then the optimal values \( {Q}_m^{\ast } \), \( {n}_1^{\ast },{n}_d^{\ast } \), and N∗ which minimize expected annual total cost for fuzzy case can be calculated by using the same previous Algorithm.