An optimization model with a lagrangian relaxation algorithm for artificial internet of things-enabled sustainable circular supply chain networks

Abstract

Circular supply chain (CSC) networks improve sustainability and create socially responsible enterprises through recycling, harvesting, and refurbishing. This study develops a Lagrangian relaxation (LR) algorithm for solving location-inventory-routing (LIR) problems with heterogeneous vehicles in multi-period and multi-product sustainable CSC networks. The proposed Artificial Internet of Things (AIoT) enabled sustainable CSC is designed to increase network performance and create a secure and traceable environment. For the first time, an LR algorithm is proposed to solve the LIR problems in an AIoT-enabled CSC network with storage, backorder shortage, split-delivery, and time window potentials. Sixteen small- and medium-size simulated problems were produced to assess the performance of the proposed algorithm relative to the GAMS software. The results show the proposed algorithm can solve the small- and medium-size problems as effectively as GAMS software but faster and more efficiently. In addition, eight large-size simulation problems were produced and solved by the algorithm. While the GAMS software failed to solve the large-size problems, the LR algorithm solved them efficiently and successfully.

Appendix

1.1 Proposed algorithm for data simulation

Parameters	Probabilistic distribution function
$FSP_{it}$	$10^{3} \times round(uniform(3 \times 10^{3} ,4 \times 10^{3} ));$
$FMN_{bjc}$	$\begin{gathered} loop(j, \hfill \\ \quad \quad \quad loop(b, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad loop(c, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if(ord(c) < = 1, \hfill \\ FMN(b,j,c) = 10^{6} \times round(uniform(1.4 \times 10^{2} ,2.1 \times 10^{2} )); \hfill \\ else \hfill \\ FMN(b,j,c) = 10^{6} \times round(1.1 \times FMN(b,j,c - 1) \times 10^{ - 6} ); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$FDST_{l}$	$10^{5} \times round(uniform(1.5 \times 10^{3} ,2.2 \times 10^{3} ));$
$FCL_{f}$	$10^{4} \times round(uniform(2.5 \times 10^{3} ,3.0 \times 10^{3} ));$
$FRC_{bdc}$	$\begin{gathered} loop(b, \hfill \\ \quad \quad \quad \quad loop(d, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad loop(c, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if(ord(c) < = 1, \hfill \\ FRC(b,d,c) = 10^{6} \times round(uniform(1.0 \times 10^{2} ,1.4 \times 10^{2} )); \hfill \\ else \hfill \\ FRC(b,d,c) = 10^{6} \times round(1.1 \times FRC(b,d,c - 1) \times 10^{ - 6} ); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$FDS_{r}$	$10^{4} \times round(uniform(4.1 \times 10^{2} ,5.0 \times 10^{2} ));$
$\begin{gathered} FSP_{i}^{IoT} \hfill \\ FMN_{j}^{IoT} \hfill \\ FDST_{l}^{IoT} \hfill \\ FCL_{f}^{IoT} \hfill \\ FRC_{d}^{IoT} \hfill \\ \end{gathered}$	$10^{3} \times round(uniform(2 \times 10^{2} ,3.0 \times 10^{2} ));$
$FVH_{h}$	$10^{5} \times round(uniform(7.5 \times 10^{2} ,8.0 \times 10^{2} ));$
$DCS_{bmt}$	$\begin{gathered} loop(b, \hfill \\ \quad \quad \quad loop(t, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad loop(m, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if(ord(m) < = 1, \hfill \\ DCS(b,m,t) = 0; \hfill \\ else \hfill \\ DCS(b,m,t) = 10^{2} \times round(uniform(9.0 \times 10^{2} ,1.0 \times 10^{3} )); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad ); \hfill \\ \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$\alpha_{bmt}$	$uniform(5.0 \times 10^{ - 2} ,9.0 \times 10^{ - 2} );$
$\omega_{ab}$	$uniform(6.0 \times 10^{ - 1} ,7.5 \times 10^{ - 1} );$
$\theta_{b}$	$uniform(2.0 \times 10^{ - 2} ,3.0 \times 10^{ - 2} );$
$EN^{IoT}$	$uniform(0.02,0.03)$
$P_{t}^{tag}$	$round(uniform(4.5,7.5));$
$PEN$	$round(uniform(5.5,6.5));$
$CPSP_{ait}$	$round(uniform(1.5,2.5) \times \frac{sum((b,m),DCS(b,m,t) \times \omega (a,b))}{{card(a) \times card(i)}})$
$\begin{gathered} CPMN_{bjc}^{L} \hfill \\ CPRC_{bdc}^{L} \hfill \\ \end{gathered}$	$0$
$CPMN_{bjc}^{U}$	$\begin{gathered} loop(j, \hfill \\ \quad \quad \quad loop(b, \hfill \\ \quad \quad \quad \quad \quad \quad \quad loop(c, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad if(ord(c) < = 1, \hfill \\ CPMNU(b,j,c) = round(uniform(1.5,2.0) \times \frac{sum((m,t),DCS(b,m,t))}{{card(j) \times card(t)}}); \hfill \\ else \hfill \\ CPMNU(b,j,c) = round(1.1 \times CPMNU(b,j,c - 1)); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$CPRC_{bdc}^{U}$	$\begin{gathered} loop(d, \hfill \\ \quad \quad \quad loop(b, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad loop(c, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if(ord(c) < = 1, \hfill \\ CPRCU(b,d,c) {=} round(uniform(7.5 {\times} 10^{ - 1} ,8.5 {\times} 10^{ - 1} ) \times \frac{sum((f,t),CPCL(b,f,t))}{{card(d) \times card(t)}}); \hfill \\ else \hfill \\ CPRCU(b,d,c) = round(1.1 \times CPRCU(b,d,c - 1)); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$CPDST_{blt}$	$round(uniform(1.5,2.0) \times \frac{sum(m,DCS(b,m,t))}{{card(l)}});$
$CPCL_{bft}$	$round(uniform(1.5,2.0) \times \frac{sum(m,DCS(b,m,t) \times \alpha (b,m,t))}{{card(f)}});$
$CPDS_{rt}$	$round(uniform(2.0 \times 10^{ - 1} ,3.0 \times 10^{ - 1} ) \times \frac{sum((b,f),CPCL(b,f,t) \times \theta (b))}{{card(r)}});$
$CPVH_{h}$	$round(uniform(1.5,1.9) \times \frac{sum((b,m,t),DCS(b,m,t) \times \theta (b))}{{card(h) \times card(t)}});$
$PDS_{at}^{RW}$	$round(uniform(8.0 \times 10^{1} ,1.0 \times 10^{2} ));$
$PDS_{bt}^{PRD}$	$round(uniform(1.1 \times 10^{2} ,1.3 \times 10^{2} ));$
$CSP_{ait}$	$round(uniform(3.0 \times 10^{3} ,3.5 \times 10^{3} ));$
$CMN_{bjt}$ $CDST_{blt}$ $CCL_{bft}$ $CRC_{bdt}$ $CDS_{brt}$	$round(uniform(2.5 \times 10^{2} ,3.5 \times 10^{2} ));$
$DMD_{jl}$ $DCC_{mf}$ $DCR_{fd}$ $DCD_{fr}$ $DRM_{dj}$	$round(uniform(2.0 \times 10^{1} ,5.0 \times 10^{1} ));$
$CHL_{bmt}$	$\begin{gathered} loop(b, \hfill \\ \quad \quad \quad \quad loop(t, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad loop(m, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if(ord(m) < = 1, \hfill \\ CHL(b,m,t) = 0; \hfill \\ else \hfill \\ CHL(b,m,t) = round(uniform(2.5 \times 10^{2} ,3.5 \times 10^{2} ))); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad ); \hfill \\ \quad \quad \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$CSH_{bmt}$	$10^{2} \times CHL(b,m,t);$
$DSM_{ij}$	$round(uniform(3.0 \times 10^{2} ,5.0 \times 10^{2} ));$
$DDC_{lm}$	$\begin{gathered} loop(l, \hfill \\ \quad \quad \quad \quad loop(m\$ (ord(m) > 1), \hfill \\ DDC(l,m) = round(uniform(2.0 \times 10^{1} ,5.0 \times 10^{1} )); \hfill \\ \quad \quad \quad ); \hfill \\ ); \hfill \\ \end{gathered}$
$TDC_{hlm}$	$round(uniform(5.0,7.0) \times DDC(l,m));$
$TC_{hmn}$	$round(uniform(5.0,7.0) \times DC(m,n));$
$\varpi_{bft}$	$uniform(0.75,0.85);$
$\mu_{h}$	$uniform(0.15,0.25);$
$PH$	$3$