Investigation of two-warehouse inventory problems in interval environment under inflation via particle swarm optimization

Figures & data

Table

$f\left(t\right)$	Demand rate dependent linearly on time t, i.e., $f\left(t\right)=a+bt$ where $a,b>0$
IFS	Inventory follows shortage
SFI	Shortage follows inventory
$S$	Highest stock level at the beginning of stock-in period
R	Highest shortage quantity due to partial backlogging
$L_1$-system	Single-warehouse system
$L_2$-system	Two-warehouse system
W	Storage capacity of OW
$\theta \left(t\right)$	Backlogging rate, which is a function of waiting time t
$\alpha ,\beta$	Deterioration rates in both warehouses OW and RW, respectively ($0<\alpha ,\beta <<1$)
r	Rate of inflation
$C_o=[C_{oL},C_{oR}]$	Interval valued inventory ordering cost per order
$C_p=[C_{pL},C_{pR}]$	Interval valued inventory purchasing cost per unit
$C_{ho}=[C_{hoL},C_{hoR}]$	Interval valued inventory holding cost per unit per unit time in OW
$C_{hr}=[C_{hrL},C_{hrR}]$	Interval valued inventory holding cost per unit per unit time in RW
$C_b=[C_{bL},C_{bR}]$	Interval valued inventory backlogging cost over the entire cycle per unit per unit time
$C_{ls}$=$[C_{lsL},C_{lsR}]$	Interval valued inventory opportunity cost due to lost sales
$TC{2}^{(1)}$,$TC{2}^{(2)}$	Present value of the total relevant costs per unit time in IFS and SFI policies, respectively, for two-warehouse inventory system
$TC{1}^{(1)}$,$TC{1}^{(2)}$	Total relevant costs per unit time in IFS and SFI policies, respectively, for single-warehouse system
$TC{W}^{(1)}$,$TC{W}^{(2)}$	Boundary costs in IFS and SFI policies, respectively, for single-warehouse system
$I_o\left(t\right),I_r(t)$	Inventory levels in OW and RW, respectively, at time t
$t_r$	Time at which the inventory level reaches zero in RW
$t_o$	Time at which the inventory level reaches zero in OW ($t_o>t_r$)
$t_s$	Time at which the shortage level reaches the lowest point in the replenishment cycle
T	Inventory cycle length
For PSO:
p_size	Population size
m_gen	Maximum number of generations
$\chi$	Constriction factor
$c_1\left(>0\right)$	Cognitive learning rate
$c_2\left(>0\right)$	Social learning rate
$r_1$, $r_2$	Uniformly distributed random numbers lying in the interval [0, 1].
${v_i}^{(k)}$	Velocity of the i-th particle at the k-th generation/iteration
${x_i}^{(k)}$	Position of the i-th particle of population at the k-th generation
$p_i^{(k)}$	Best previous position of the i-th particle at the k-th generation
$p_g^{(k)}$	Position of the best particle among all the particles in the population

Figure 1. Pictorial representation of two-warehouse inventory system with SFI policy.

Table 1. Best found solution of the model with IFS policy by PSO-CO.

$f(t)$	$\theta (t_s-t)$	$t_r$	$t_o$	$t_s$	$S^{\ast }$	$R^{\ast }$	Average cost	Centre value of Average cost	Action
$a+bt$	$\frac{1}{1+\delta (t_s-t)}$	0.4057	0.6446	0.7091	264.6311	25.9580	[170.0589,406.2621]	288.1694	$L_2$-system
$a+bt$	$e^{-\delta (t_s-t)}$	0.4116	0.6503	0.7129	266.9497	25.2167	[169.4168, 406.9106]	288.1637	$L_2$-system

Table 2. Best found solution of the model with IFS policy by GQPSO.

$f(t)$	$\theta (t_s-t)$	$t_r$	$t_o$	$t_s$	$S^{\ast }$	$R^{\ast }$	Average cost	Centre value of Average cost	Action
$a+bt$	$\frac{1}{1+\delta (t_s-t)}$	0.4113	0.65	0.7135	266.8377	25.562	[169.4372, 406.6552]	288.0462	$L_2$-system
$a+bt$	$e^{-\delta (t_s-t)}$	0.4116	0.6503	0.7129	266.9496	25.2167	[169.4168, 406.9106]	288.1637	$L_2$-system

Table 3. Best found solution of the model with IFS policy by WQPSO.

$f(t)$	$\theta (t_s-t)$	$t_r$	$t_o$	$t_s$	$S^{\ast }$	$R^{\ast }$	Average cost	Centre value of Average cost	Action
$a+bt$	$\frac{1}{1+\delta (t_s-t)}$	0.4113	0.6500	0.71351	266.8333	25.5626	[169.4381, 406.65427]	288.0462	$L_2$-system
$a+bt$	$e^{-\delta (t_s-t)}$	0.4116	0.6503	0.7129	266.94843	25.2167	[169.4171, 406.9103]	288.1637	$L_2$-system

Table 4. Best found solution of the model with SFI policy by PSO-CO.

$f(t)$	$\theta (t_s-t)$	$t_s$	$t_r$	$t_o$	$S^{\ast }$	$R^{\ast }$	Average cost	Centre value of Average cost	Action
$a+bt$	$\frac{1}{1+\delta (t_s-t)}$	0.0632	0.4776	0.7172	268.4868	24.8577	[169.0776,406.2361]	287.6567	$L_2$-system
$a+bt$	$e^{-\delta (t_s-t)}$	0.0624	0.4770	0.7166	268.5872	24.523	[169.0563, 406.4916]	287.7740	$L_2$-system

Table 5. Best found solution of the model with SFI policy by GQPSO.

$f(t)$	$\theta (t_s-t)$	$t_s$	$t_r$	$t_o$	$S^{\ast }$	$R^{\ast }$	Average cost	Centre value of Average cost	Action
$a+bt$	$\frac{1}{1+\delta (t_s-t)}$	0.0632	0.4776	0.7172	268.4867	24.8577	[169.0772, 406.2361]	287.6567	$L_2$-system
$a+bt$	$e^{-\delta (t_s-t)}$	0.0624	0.4770	0.7166	268.5872	24.5228	[169.0563, 406.4916]	287.7740	$L_2$-system

Table 6. Best found solution of the model with SFI policy by WQPSO.

$f(t)$	$\theta (t_s-t)$	$t_s$	$t_r$	$t_o$	$S^{\ast }$	$R^{\ast }$	Average cost	Centre value of Average cost	Action
$a+bt$	$\frac{1}{1+\delta (t_s-t)}$	0.0632	0.4776	0.7172	268.4873	24.85776	[169.0771, 406.2362]	287.6567	$L_2$-system
$a+bt$	$e^{-\delta (t_s-t)}$	0.0624	0.477	0.7166	268.5860	24.5231	[169.0565, 406.4913]	287.7740	$L_2$-system

Table 7. Statistical analysis of the results obtained for different demand and backlogging rates in SFI by PSO-CO.

	$\theta (t_s-t)=\frac{1}{1+\delta (t_s-t)}$	$\theta (t_s-t)=e^{-\delta (t_s-t)}$
Mean objective value	[169.0776, 406.2361]	[169.0563, 406.4916]
S.D. of centre objective value*	1.1649 × 10^–13	5.8247 × 10^–14
C.O.V. of centre objective value*	4.0497 × 10^–16	2.0240 × 10^–16
S.D. of lower bound	2.9123 × 10^–14	5.8247 × 10^–14
S.D. of upper bound	4.9761 × 10^–05	5.8247 × 10^–14
C.O.V. of lower bound	1.7225 × 10^–16	3.4454 × 10^–16
C.O.V. of upper bound	1.2249 × 10^–07	1.4329 × 10^–16

Table 8. Statistical analysis of the results obtained for different demand and backlogging rates in SFI by GQPSO.

	$\theta (t_s-t)=\frac{1}{1+\delta (t_s-t)}$	$\theta (t_s-t)=e^{-\delta (t_s-t)}$
Mean objective value	[169.0772, 406.2361]	[169.0563, 406.4916]
S.D. of centre objective value*	1.1664 × 10^–13	0.0000
C.O.V. of centre objective value*	4.0500 × 10^–16	0.0000
S.D. of lower bound	2.9160 × 10^–14	5.8320 × 10^–14
S.D. of upper bound	4.4426 × 10^–05	1.1664 × 10^–13
C.O.V. of lower bound	1.7200 × 10^–16	3.4500 × 10^–16
C.O.V. of upper bound	1.0900 × 10^–07	2.8700 × 10^–16

Table 9. Statistical analysis of the results obtained for different demand and backlogging rates in SFI by WQPSO.

	$\theta (t_s-t)=\frac{1}{1+\delta (t_s-t)}$	$\theta (t_s-t)=e^{-\delta (t_s-t)}$
Mean objective value	[169.0771, 406.2362]	[169.0565, 406.4913]
S.D. of centre objective value*	1.1664 × 10^–13	0.0000
C.O.V. of centre objective value*	4.0500 × 10^–16	0.0000
S.D. of lower bound	0.0006	0.0005
S.D. of upper bound	0.0006	0.0005
C.O.V. of lower bound	3.6300 × 10^–06	3.4400 × 10^–06
C.O.V. of upper bound	1.4900 × 10^–06	1.4200 × 10^–06

Figure 2. Percentage changes in centre value of average cost w.r.t W, b,$\beta$ and r.

Figure 3. Percentage changes in centre value of average cost w.r.t a and $\alpha$.

Figure 4. Percentage changes in S w.r.t W, b, $\alpha$and r.

Figure 5. Percentage changes in S w.r.t a and $\beta$.

Figure 6. Percentage changes in R w.r.t W, b, $\beta$ and r.

Figure 7. Percentage changes in R w.r.t a and $\alpha$.

Figure A.1. Pictorial representation of two-warehouse inventory system with IFS policy.