Coordination of production planning and distribution in closed-loop supply chains

Abstract

A closed-loop supply chain structure organises material and information flows from origin points to consumption points, including production, recycling, disposal, and other reverse logistic activities. Some integration problems arise with this structure including production, inventory, location, routing, distribution, collection, recycling, and routing. The integration problems that are facing scientific researchers include inventory routing, location routing, and location inventory. This study considers the integration problem of a closed-loop supply chain for the production, distribution, collection, and recycling quantities, along with the distribution and collection routes for each time period of a finite planning horizon. We refer to this problem as the “Closed-Loop Supply Chain Integrated Production-Inventory-Distribution-Routing Problem” (CLSC-PRP). A mathematical model is proposed that is the first to determine both quantities and routes for the CLSC-PRP simultaneously. As the problem is known to be NP-hard in terms of computational complexity, a simulated annealing-based decomposition heuristic is developed for solving large-scale CLSC-PRP instances. The results of the proposed mathematical model for the CLSC-PRP are compared with the results of the developed heuristic and two separate models that manage forward and backward production routing problems. An extensive comparative study indicated the following: (i) the proposed model was able to reduce the cost required for operating the total supply chain by an average of 12%, along with providing a positive impact on the environment and (ii) the proposed heuristic is able to generate solutions that are close to optimal in most cases.

Appendices

Appendix 1

Notation of Mathematical Formulations

Indices

$i,j,v$ :

${\text{Indices}}\;{\text{for}}\;{\text{nodes,}}\;{\text{where}}\; 0\;{\text{denotes}}\;{\text{the}}\;{\text{facility,}}\;{\text{recyler,}}\;{\text{and}}\;{\text{warehouse}}$

$k$ :

${\text{Index}}\;{\text{for}}\;{\text{vehicles}}$

$t$ :

${\text{Index}}\;{\text{for}}\;{\text{periods}}$

$N_{c}$ :

${\text{Set}}\;{\text{of}}\;{\text{consumption}}\;{\text{nodes}}\;N = N_{c} \cup \left\{ 0 \right\}$

$K$ :

${\text{Set}}\;{\text{of}}\;{\text{avaliable}}\;{\text{trucks}}$

$T$ :

${\text{Set}}\;{\text{of}}\;{\text{periods}}\;{\text{in}}\;{\text{the}}\;{\text{planning}}\;{\text{horizon}}\;T_{0} = T \cup \left\{ 0 \right\}$

Parameters

$C_{k}$ :

${\text{Capacity}}\;{\text{of}}\;{\text{vehicle}}\;k\;\left( {\text{unit}} \right)$

$B ^{m} \left( {B ^{r} } \right)$ :

${\text{Manufacturing}}\;\left( {\text{recycling}} \right)\;{\text{capacity}}\;\left( {\text{unit/period}} \right)$

$q_{t}^{m} \left( {w_{t}^{r} } \right)$ :

${\text{Raw}}\;{\text{material}}\;{\text{purchasing}}\;\left( {{\text{returned}}\;{\text{product}}\;{\text{collection}}} \right)\;{\text{cost}}\;{\text{for}}\;{\text{period}}\;t \;\left( {\text{money/unit}} \right)$

$g_{i}^{m} \left( {g_{i}^{r} } \right)$ :

${\text{Minimum }}\;{\text{new}}\;\left( {\text{returned}} \right)\;{\text{inventory}}\;{\text{level}}\;{\text{of}}\;{\text{node}}\;i\;\left( {\text{unit}} \right)$

$a_{i}^{m} \left( {a_{i}^{r} } \right)$ :

${\text{Maximum}}\;{\text{new}}\;\left( {\text{returned}} \right)\;{\text{inventory}}\;{\text{level}}\;{\text{of}}\;{\text{node}}\;i\;\left( {\text{unit}} \right)$

$f_{k}$ :

${\text{Fixed}}\;{\text{usage}}\;{\text{cost}}\;{\text{for}}\;{\text{vehicle}}\;k \;\left( {\text{money/period}} \right)$

$c_{{}}^{v}$ :

${\text{Variable}}\;{\text{cost}}\;{\text{for}}\;{\text{transportation}}\;\left( {\text{money/distance}} \right)$

$s_{t}^{m} \left( {s_{t}^{r} } \right)$ :

${\text{Production}}\;\left( {\text{recycling}} \right)\;{\text{setup }}\;{\text{cost}}\;{\text{for}}\;{\text{period }}\;t\; \left( {\text{money/period}} \right)$

$c_{t}^{m} \left( {c_{t}^{r} } \right)$ :

${\text{Variable}}\;{\text{production}}\;\left( {\text{recycling}} \right)\;{\text{cost}}\;{\text{for}}\;{\text{period}}\;t\; \left( {\text{money/period}} \right)$

$h_{i}^{m} \left( {h_{i}^{m} } \right)$ :

${\text{Holding}}\;{\text{cost}}\;{\text{for}}\;{\text{new}}\;\left( {\text{returned}} \right)\;{\text{product}}\;{\text{of}}\;{\text{node}}\;i\; ( {\text{money/}}\left( {{\text{unit}} \times {\text{period}}} \right)$

$l_{ij}$ :

${\text{Distance}}\;{\text{between }}\left( {i,j} \right)\;{\text{pair}}\;\left( {\text{distance}} \right)$

$L$ :

${\text{Maximum tour length in a period}}$

$d_{it}^{m} \left( {d_{it}^{r} } \right)$ :

${\text{New }}\left( {\text{returned}} \right)\,{\text{product demand of node }}i\; {\text{at period }}t \left( {\text{unit}} \right)$

$\alpha$ :

${\text{production rate}}$

$\beta$ :

${\text{Maximum }}\;{\text{recycling }}\;{\text{ratio}}$

$\gamma$ :

${\text{Coefficient}}\;{\text{of}}\;{\text{calculating}}\;{\text{the}}\;{\text{returned}}\;{\text{product}}\;{\text{regarding}}\;{\text{recycling}}\;{\text{quantities}}$

Decision Variables

$p_{t} \left( {p_{t}^{r} } \right)$ :

${\text{Production }}\left( {\text{recycling}} \right) {\text{quantity at period }}t\; \left( {\text{unit}} \right)$

$p_{t}^{m}$ :

${\text{Purshased}}\;{\text{raw}}\;{\text{material}}\;{\text{quantity}}\;{\text{at}}\; {\text{period }}\;t \;\left( {\text{unit}} \right)$

$Y _{t}^{m} \left( {Y _{t}^{r} } \right)$ :

$\left\{ {\begin{array}{*{20}l} {1,} \hfill & {{\text{if}}\;{\text{new}}\;\left( {{\text{returned}}} \right)\;{\text{product}}\;{\text{is}}\;{\text{produced}}\;\left( {{\text{recycled}}} \right)\; }\\ &{ {\text{at}}\;{\text{period}}\;t} \hfill \\ {0,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$

$I_{it}^{m} \left( {I_{it}^{r} } \right)$ :

${\text{New }}\;\left( {\text{returned}} \right) \;{\text{inventory}}\; {\text{level}}\; {\text{on}}\; {\text{node}}\; i\; {\text{at}}\; {\text{period }}\;t \;\left( {\text{unit}} \right)$

$Q_{ikt}^{m} \left( {Q_{ikt}^{r} } \right)$ :

${\text{New }}\left( {\text{returned}} \right)\,\,{\text{product delivery }}\left( {\text{pick up}} \right) {\text{amount to }}\left( {\text{from}} \right)\;{\text{node }}i {\text{at period }}t {\text{with vehicle }}k \left( {\text{unit}} \right)$

$X_{ijkt}^{m} \left( {X_{ijkt}^{r} } \right)$ :

$N{\text{ew }}\left( {\text{returned}} \right) {\text{product load on travelling }}\left( {i,j} \right) {\text{pair at period }}t\;{\text{with vehicle }}k\;\left( {\text{unit}} \right)$

$Z_{\text{ijkt}}$ :

$\left\{ {\begin{array}{*{20}l} {{\text{1,}}} \hfill & {{\text{if~}}\;{\text{a~}}\;{\text{delivery~}}\;{\text{made}}\;{\text{~to}}\;{\text{~}}\left( {i,j} \right)\;{\text{~pair~}}\;{\text{at}}\;{\text{~period}}\;{\text{~}}t\; } \\ & { {\text{~with~}}\;{\text{vehicle}}\;{\text{~}}k} \hfill \\ {{\text{0,}}} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$

$U_{\text{kt}}$ :

$\left\{ {\begin{array}{*{20}l} { 1 , } \hfill & {{\text{if vehicle }}k\,\,{\text{is used on period }}t} \hfill \\ { 0 ,} \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.$

Appendix 2

Parameter setting case problem instanceIn this parameters setting test instance, a 7-day planning problem is considered, and the dataset is briefly described in Table 9. Distribution activities are carried out by different capacities of vehicles (72.09 ton), and during the planning horizon, eight different points demand new products or collection requests for recycling. The distances between cities are generated randomly.

Table 9 Parameters of the problem instance