Reliability Dependent Imperfect Production Inventory Optimal Control Fractional Order Model for Uncertain Environment Under Granular Differentiability

Figures & data

Figure 1. The μ-level set of selling price for $\text{β}=0.8$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of selling price $P_u\left(t\right)$ and the bottom line indicates the lower level of selling price $P_l\left(t\right)$.

Table 1. Values of unchangeable parameter.

Parameters	Values	Parameters	Values	Parameters	Values	Parameters	Values	Parameters	Values
$a_1$	0.65	$a_2$	0.05	$a_3$	0.05	α	0.01	${\alpha }_{d_0}$	0.7
${\alpha }_u$	0.8	${\alpha }_q$	0.07	${\alpha }_{\lambda }$	0.85	${\alpha }_{cu0}$	0.8	${\alpha }_{cu1}$	0.75
${\alpha }_{cu2}$	0.75	${\alpha }_{ps}$	0.85	${\alpha }_h$	0.9	${\alpha }_r$	0.9	${\alpha }_p$	0.75

Table 2. Input data of fuzzy parameters.

Parameters	Fuzzy values	Horizontal membership function
${\tilde{d}}_0$	$(35,40,45)$	$d_0^{gr}=35+5\mu +10(1-\mu ){\alpha }_{d_0}$
${\tilde{C}}_{u0}$	$(0.1,0.3,0.5)$	$C_{u0}^{gr}=0.1+0.2\mu +0.4(1-\mu ){\alpha }_{cu0}$
${\tilde{C}}_{u2}$	$(0.2,0.4,0.6)$	$C_{u2}^{gr}=0.2+0.2\mu +0.4(1-\mu ){\alpha }_{cu2}$
$\tilde{\lambda }$	$(20,20.2,20.4)$	${\lambda }^{gr}=20+0.2\mu +0.4(1-\mu ){\alpha }_{\lambda }$
$\tilde{r}$	$(0.85,0.9,0.96)$	$r^{gr}=0.85+0.05\mu +0.11(1-\mu ){\alpha }_r$
${\tilde{C}}_{u1}$	$(2,4,6)$	$C_{u1}^{gr}=2+2\mu +4(1-\mu ){\alpha }_{cu1}$
$\tilde{U}$	$(20,22,24)$	$U^{gr}=20+2\mu +4(1-\mu ){\alpha }_u$
$\tilde{Q}$	$(25,27,29)$	$Q^{gr}=25+2\mu +4(1-\mu ){\alpha }_q$
$\tilde{h}$	$(0.2,0.5,0.8)$	$h^{gr}=0.2+0.3\mu +0.6(1-\mu ){\alpha }_h$
${\tilde{P}}_s$	$(30,35,40)$	$P_s^{gr}=30+5\mu +10(1-\mu ){\alpha }_{ps}$

Table 3. The value of selling price, stock level of product, production rate, demand rate, profit at time T for $\text{β}=0.8$.

t	0	10	20	30	40	50	60	70	80	90	100
$P_u^{\ast }$	43.47	43.39	43.33	43.27	43.20	43.13	43.07	43	42.93	42.86	42.80
$P_l^{\ast }$	42.70	42.62	42.56	42.49	42.43	42.36	42.29	42.23	42.16	42.09	42.02
$X_u^{\ast }$	0	0.3406	0.3847	0.3648	0.3316	0.2959	0.2602	0.2250	0.1905	0.1565	0.1232
$X_l^{\ast }$	0	0.2368	0.2605	0.2383	0.2066	0.1733	0.1402	0.1075	0.0754	0.0440	0.0131
$U_u^{\ast }$	22.34	21.94	21.57	21.20	20.83	20.47	20.11	19.76	19.41	19.06	18.72
$U_l^{\ast }$	20.55	20.19	19.85	19.51	19.17	18.84	18.51	18.18	17.86	17.54	17.22
$D_u^{\ast }$	15.14	15.20	15.24	15.29	15.33	15.37	15.41	15.45	15.49	15.53	15.57
$D_l^{\ast }$	14.67	14.73	14.77	14.82	14.86	14.90	14.94	14.98	15.02	15.06	15.10
$J_u^{\ast }$	443.1	451.2	458.9	466.3	473.5	480.5	487.4	494	500.5	506.9	513.1
$J_l^{\ast }$	398.3	407.1	415.4	423.3	431.1	438.6	446	453.1	460.1	466.9	473.5

Table 4. The value of selling price, stock level of product, production rate, demand rate, profit at time T for $\text{β}=0.9$.

t	0	10	20	30	40	50	60	70	80	90	100
$P_u^{\ast }$	43.47	43.39	43.33	43.27	43.20	43.13	43.07	43	42.93	42.86	42.80
$P_l^{\ast }$	42.70	42.62	42.56	42.49	42.43	42.36	42.29	42.23	42.16	42.09	42.02
$X_u^{\ast }$	0	0.509	0.7254	0.7927	0.7842	0.7375	0.6721	0.5977	0.5195	0.4401	0.3609
$X_l^{\ast }$	0	0.371	0.495	0.5265	0.5027	0.4513	0.3863	0.3152	0.2415	0.1673	0.0934
$U_u^{\ast }$	22.34	21.94	21.57	21.20	20.83	20.47	20.11	19.76	19.41	19.06	18.72
$U_l^{\ast }$	20.55	20.19	19.85	19.51	19.17	18.84	18.51	18.18	17.86	17.54	17.22
$D_u^{\ast }$	15.14	15.20	15.24	15.29	15.33	15.37	15.41	15.45	15.49	15.53	15.57
$D_l^{\ast }$	14.67	14.73	14.77	14.82	14.86	14.90	14.94	14.98	15.02	15.06	15.10
$U_u^{\ast }$	446.3	454.6	462.4	469.9	477.1	484.1	490.9	497.6	504	510.3	516.4
$U_l^{\ast }$	407.2	416.30	424.7	432.8	440.5	448.1	455.4	462.5	469.4	476.2	482.7

Table 5. The value of selling price, stock level of product, production rate, demand rate, profit at time T for $\text{β}=1$.

t	0	10	20	30	40	50	60	70	80	90	100
$P_u^{\ast }$	43.47	43.38	43.33	43.27	43.22	43.16	43.10	43.04	42.98	42.92	42.85
$P_l^{\ast }$	42.70	42.62	42.58	42.54	42.49	42.44	42.39	42.34	42.28	42.23	42.17
$X_u^{\ast }$	0	0.7236	1.381	1.975	2.507	2.977	3.387	3.738	4.031	4.268	4.449
$X_l^{\ast }$	0	0.5002	0.9863	1.321	1.644	1.911	2.123	2.28	2.384	2.436	2.438
$U_u^{\ast }$	22.34	21.94	21.57	21.20	20.83	20.47	20.11	19.76	19.41	19.06	18.72
$U_l^{\ast }$	20.55	20.19	19.85	19.51	19.17	18.84	18.51	18.18	17.86	17.54	17.22
$D_u$	15.14	15.20	15.25	15.31	15.36	15.41	15.45	15.50	15.55	15.59	15.63
$D_l^{\ast }$	14.67	14.73	14.79	14.85	14.90	14.96	15.01	15.06	15.11	15.16	15.21
$U_u^{\ast }$	557	566.3	575.3	584	592.4	600.6	608.5	616.1	623.5	630.6	637.5
$U_l^{\ast }$	536.4	544.9	553	561	568.6	576	583.2	590.1	596.8	603.2	609.5

Figure 2. The μ-level set of stock level for $\text{β}=0.8$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of stock level $X_u\left(t\right)$ and the bottom line indicates the lower level of stock level $X_l\left(t\right)$.

Figure 3. The μ-level set of unit production for $\text{β}=0.8$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of unit production $U_u\left(t\right)$ and the bottom line indicates the lower level of unit production $U_l\left(t\right)$.

Figure 4. The μ-level set of demand for $\text{β}=0.8$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of demand $D_u\left(t\right)$ and the bottom line indicates the lower level of demand $D_l\left(t\right)$.

Figure 5. The μ-level set of total profit for $\text{β}=0.8$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of total profit $J_u\left(t\right)$ and the bottom line indicates the lower level of total profit $J_l\left(t\right)$.

Figure 6. The μ-level set of selling price for $\text{β}=0.9$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of selling price $P_u\left(t\right)$ and the bottom line indicates the lower level of selling price $P_l\left(t\right)$.

Figure 7. The μ-level set of stock level for $\text{β}=0.9$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of stock level $X_u\left(t\right)$ and the bottom line indicates the lower level of stock level $X_l\left(t\right)$.

Figure 8. The μ-level set of unit production for $\text{β}=0.9$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of unit production $U_u\left(t\right)$ and the bottom line indicates the lower level of unit production $U_l\left(t\right)$.

Figure 9. The μ-level set of demand for $\text{β}=0.9$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of demand $D_u\left(t\right)$ and the bottom line indicates the lower level of demand $D_l\left(t\right)$.

Figure 10. The μ-level set of total profit for $\text{β}=0.9$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of total profit $J_u\left(t\right)$ and the bottom line indicates the lower level of total profit $J_l\left(t\right)$.

Figure 11. The μ-level set of selling price for $\text{β}=1$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of selling price $P_u\left(t\right)$ and the bottom line indicates the lower level of selling price $P_l\left(t\right)$.

Figure 12. The μ-level set of stock level for $\text{β}=1$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of stock level $X_u\left(t\right)$ and the bottom line indicates the lower level of stock level $X_l\left(t\right)$.

Figure 13. The μ-level set of unit production for $\text{β}=1$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of unit production $U_u\left(t\right)$ and the bottom line indicates the lower level of unit production $U_l\left(t\right)$.

Figure 14. The μ-level set of demand for $\text{β}=1$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of demand $D_u\left(t\right)$ and the bottom line indicates the lower level of demand $D_l\left(t\right)$.

Figure 15. The μ-level set of total profit for $\text{β}=1$. The ten curves show left and right end points of μ-level sets of fuzzy function respectively, for $\mu \in [0,1]$. The top most line indicates the upper level of total profit $J_u\left(t\right)$ and the bottom line indicates the lower level of total profit $J_l\left(t\right)$.