A system dynamics model for the impact of capacity limits on the Bullwhip effect (BWE) in a closed-loop system with remanufacturing


Every organisation has an upper limit to the number of orders or products that it can manufacture or remanufacture per unit time. In reverse logistics operations, capacity limits can lead to inefficiencies in the remanufacturing process. In this paper, comparisons were made of the Bullwhip effect (BWE) in closed-loop systems that have collection and remanufacturing capacity limits and those that do not. Collection and remanufacturing capacity limits were introduced for a system where a company had to collect ‘enough’ products before remanufacturing can begin. This introduced collection backlogs, remanufacturing backlogs and remanufacturing downtimes to the closed loop supply chain (CLSC). By adopting a systems dynamics approach, the research performed ‘what-if’ analyses of the closed-loop system under different levels of the factors under investigation. Two case studies were investigated: one remanufacturing electric vehicle batteries (low demand, slow moving item) and the other remanufacturing kitchen appliances (high demand, fast moving item). Firstly, introducing collection and remanufacturing capacity limits in the reverse chain increased the BWE to a level higher than the reverse chain without any capacity limits, but not to the level of the forward chain without any product returns. Secondly, introducing collection and remanufacturing capacity limits for a closed-loop system where a company had to collect ‘enough’ products before remanufacturing begins had different impacts depending on the product demand size and speed. The presence of external returns by other parties not regulated by an organisation had an impact of lowering the BWE in the closed-loop system and it also impacted how the other factors under investigation affected the Bullwhip effect. These findings were used to provide managerial insights for organisations venturing into reverse logistics.

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Stock and flow diagrams and model equations

Fig. 10

Stock and flow diagram for the serial forward chain in STELLA

Fig. 11

Stock and flow diagram for the reverse chain without some components of the forward chain in STELLA

Model equations


  1. 1.

    Collected products

    $$ Collected\ products(t)= collected\ products\left(t- dt\right)+\left( total\ collection\ rate- acce\ rate\ for\ reuse- reje\ rate\ for\ reuse\right)\times dt $$
$$ Initial\ collected\ products=0 $$


$$ total\ collection\ rate=\mathit{\operatorname{MIN}}\left( compan{y}^{\prime } scollection\ capacity, available\ used\ products+ products\ not\ collected\right) $$


$$ acce\ rate\ for\ prod\ reuse= collected\ products\times \frac{reuse\%}{ reuse\ inspe\ time} $$
$$ reje\ rate\ for\ prod\ reuse=\left(1- reuse\%\right)\times \frac{collected\ products}{reuse\ insp\ time} $$
  1. 2.

    Components inventory

    $$ Components\ inventory\ (t)= components\ inventory\ \left(t- dt\right)+\left( comp\ prdn\ rate+ comp\ reman\ rate+ comp\ acce\ rate\ for\ dir\ reuse- comp\ used\ for\ prdn\right)\times dt $$
$$ Initial\ components\ inventory=0 $$


$$ comp\ prdn\ rate=\mathit{\operatorname{MAX}}\left(\mathit{\operatorname{MIN}}\left(\mathit{\operatorname{MIN}}\left(\frac{raw\ mat\ inventory}{comp\ prod\ time},\left( expe\ distributor\ orders\times comp\ per\ product- expe\ reusable\ compo+\frac{CI\ discrep}{CI\ adj\ time}\right), comp\ prod\ capacity\right),0\right)\right) $$
$$ comp\ reman\ rate=\mathit{\operatorname{MIN}}\left( comp\ reman\ capacity,\frac{\left(1- disposal\%\right)\times comp\ reje\ for\ dir\ reuse}{reproce\ time}\right) $$
$$ comp o\ acceptance\ rate\ for\ dir\ reuse=\frac{dir\ reuse\%\times inv\ of\ comp\ from\ reje\ prdcts}{insp\ and\ disasse\ time} $$


$$ comp\ used\ for\ prdn=\mathit{\operatorname{MAX}}\left(\mathit{\operatorname{MIN}}\left(\mathit{\operatorname{MIN}}\left(\frac{compo\ inventory}{prdct\ prdn\ time}, prdn\ capacity\times comp\ per\ prdt\right),\left( expe\ dist\ orders- expe\ reman\ prdcts+\frac{SI\ discrep}{SI\ adj\ time}\right)\times comp\ per\ prdt\right),0\right) $$
  1. 3.

    Components rejected for direct reuse

    $$ comp\ reje\ for\ dir\ reuse\ (t)= comp\ reje\ for\ dir\ reuse\ \left(t- dt\right)+\left( comp\ repla\ rate+ comp\ reje\ rate\ for\ dir\ reuse- recycle\ rate- coomp\ reman\ rate- disposal\ rate\right)\times dt $$
$$ initial\ comp\ reje\ for\ dir\ reuse=0 $$


$$ comp\ repla\ rate= reman\ rate\times comp\ per\ prdct\times comp\ repla\% comp\ reje\ rate\ for\ dir\ reuse=\frac{\left(1- dir\ reuse\%\right)\times inv\ of\ comp\ from\ reje\ prdcts}{insp\ and\ disasse\ time} $$
$$ comp\ reje\ rate\ for\ dir\ reuse=\frac{\left(1- dir\ reuse\%\right)\times inv\ of\ comp\ from\ reje\ prdcts}{insp\ and\ disasse\ time} $$


$$ recycling\ rate=\frac{comp\ reje\ for\ dir\ reuse}{reproce\ time} $$
$$ comp\ reman\ rate=\frac{\left(1- disposal\%\right)\times comp\ reje\ for\ dir\ reuse}{reproce\ time} $$
$$ disposal\ rate= comp\ reje\ for\ dire\ reuse\times disposal\% $$
  1. 4.

    Controllable disposal

    $$ controllable\ disposal\ (t)= controllable\ disposal\ \left(t- dt\right)+\left( disposal\ rate\right)\times dt $$
$$ initial\ controllable\ disposal=0 $$


$$ disp osal\ rate= comp\ reje\ for\ dir\ reuse\times disp\% $$
  1. 5.

    Distributor orders backlog

    $$ distributor\ orders\ backlog\ (t)= distributor\ orders\ backlog\ \left(t- dt\right)+\left( distributor\ orders- distributor\kern0.5em backlog\ red\ rate\right)\times dt $$
$$ initial\ distributor\ orders\ backlog=0 $$


$$ distributors\ orders= expe\ wholesale\ orders+\frac{DI\ discrepancy}{DI\ adj\ time} $$


$$ distributors\ backlog\ red\ rate= shipments\ to\ distributor $$
  1. 6.

    Distributor inventory

    $$ distributor\ inventory\ (t)= distributor\ inventory\ \left(t- dt\right)+\Big( shipments\ to\ distributor- shipments\ to\ wholesaler\times dt $$
$$ initial\ distributor\ inventory=0 $$


$$ {\displaystyle \begin{array}{l} shipments\ to\ distributor=\\ {}\frac{\left( IF\left( serviceable\ inventory- distributors\ orders\ backlog\ge 0\right) THEN\left( distributors\ orders\ backlog\right) ELSE\left( serviceable\ inventory\right)\right)}{distr\ shipment\ time}\end{array}} $$


$$ shipment s\ to\ wholesaler=\frac{\left( IF\left( distributor\ inventory- wholesale\ orders\ backlog\ge 0\right) THEN\left( wholesale\ orders\ backlog\right) ELSE\left( distributor\ inventory\right)\right)}{wholesale\ shipment\ time} $$
  1. 7.

    Inventory of components from rejected products

    $$ inv\ of\ comp\ from\ reje\ prdcts\ (t)= inv\_ of\_ comp\_ from\_ reje\_ prdcts\left(t- dt\right)+\left( comp\_ from\_ reje\_ prodcts- comp\_ reje\_ rate\_ for\_ dir\_ reuse- comp o\_ acce\_ rate\_ for\_ dir ect\_ reuse\right)\times dt $$
$$ initial\ inventory\ of\ comp\ from\ rejected\ products=0 $$


$$ comp\ from\ reje\ products= reje\_\_ rate\_ for\_ reman\times comp onents\_ per\_ product $$


$$ comp\ reje\ rate\ for\ dir\ reuse=\frac{\left(1- direct\_ reuse\_\%\right)\times inv\_ of\_ comp\_ from\_ reje\_ prdcts}{inspe\ and\ dissasse\ time} $$
$$ compo\ acce\ for\ dire\ reuse=\frac{\left( direct\_ reuse\_\%\times inv\_ of\_ comp\_ from\_ reje\_ prdcts\right)}{insp\ and\ dissasse\ time} $$
  1. 8.

    Products for remanufacturing

    $$ products\ for\ remn\ (t)= products\ for\ reman\ \left(t- dt\right)+\left( acce\ rate\ for\ reman- reman\ rate- produsctscotremanufactured\right)\times dt $$
$$ initial\ products\ for\ reman=0 $$


$$ acce\ rate\ for\ reman=\frac{reje\ prod\ for\ reuse\times reman\%}{initial\ inspe\ time} $$


$$ reman\ rate= IF\left(\frac{products\ for\ reman+ reman\ backlogs}{reprocess\ time}\ge reman\ capacity\right) THEN\left( reman\ capacity\right) ELSE(0) $$
$$ products\ not\ reman ufactured=\mathit{\operatorname{MAX}}\left(\left(\frac{products\ for\ reman}{reprocess\ time}\right)- reman\ capacity,0\right) $$
  1. 9.

    Products for cannibalisation

    $$ prod\ for\ cannibalisation\ (t)= prod\ for\ cannibalisation\ \left(t- dt\right)+\left( reje\ rate\ for\ reman\right)\times dt $$
$$ initial\ products\ for\ cannibalisation=0 $$


$$ reje\ rate\ for\ reman=\frac{\left(1- reman\%\right)\times reje\ prod\ for\ reuse}{initial\ inspe\ time} $$
  1. 10.

    Raw material inventory

    $$ raw\ mat\ inventory\ (t)= raw\ mat\ inventory\ \left(t- dt\right)+\left( recycling\ rate- comp\ prod\ rate\right)\times dt $$
$$ initial\ raw\ mat\ inventory= infinity $$
$$ recycling\ rate=\frac{comp\ reje\ for\ dir\ reuse}{reproce\ time} $$


$$ comp\ prdn\ rate=\mathit{\operatorname{MAX}}\left(\mathit{\operatorname{MIN}}\left(\mathit{\operatorname{MIN}}\left(\frac{raw\ mat\ inventory}{comp\ prod\ time},\left(\left( expe\ distri\ orders\times comp\ per\ prdct\right)- expe\ reusable\ compo+\frac{CI\ discrepe}{CI\ adj\ time}\right)\right), comp\ prod\ capacity\right),0\right) $$
  1. 11.

    Rejected products for reuse

    $$ reje\ prod\ for\ reuse\ (t)= reje\ prod\ for\ reuse\ \left(t- dt\right)+\left( reje\ rate\ for\ prod\ reuse- acce\ rate\ for\ reman- reje\ rate\ for\ reman\right)\times dt $$
$$ initial\ reje\ prod\ for\ reuse=0 $$


$$ reje\ rate\ for\ prod\ reuse=\frac{\left(1- reuse\%\right)\times collected\ products}{reuse\ insp\ time} $$


$$ acce\ rate\ for\ reman=\frac{reje\ prod\ for\ reuse\times reman\%}{initial\ inspe\ time} $$
$$ reje\ rate\ for\ reman=\frac{\left(1- reman\%\right)\times reje\ prod\ for\ reuse}{initial\ inspe\ time} $$
  1. 12.

    Remanufacturing backlogs

    $$ remanufacturing\ backlogs\ (t)= remanufacturing\ backlogs\ \left(t- dt\right)+\left( products\ not\ remanufactured\right)\times dt $$
$$ initial\ remanufacturing\ backlogs=0 $$


$$ products\ not\ reman ufactured=\mathit{\operatorname{MAX}}\left(\left(\left(\frac{products\ for\ reman}{reprocess\ time}\right)- reman\ capacity\right),0\right) $$
  1. 13.

    Retailer orders backlog

    $$ retaile{r}^{\prime } s order\ backlog\ (t)= retailer\ orders\ backlog\ \left(t- dt\right)+\left( retaile{r}^{\prime } s order s- retailer\ backlog\ red\ rate\right)\times dt $$
$$ initial\ retailer\ orders\ backlog=0 $$


$$ retaile{r}^{\prime } sorders= expected\ demand+\left(\frac{RI\ discrepency}{RI\ adj\ time}\right) $$


$$ retailer\ backlog\ redu\ rate= shipments\ to\ retailer $$
  1. 14.

    Retailer inventory

    $$ retailer\ inventory\ (t)= retailer\ inventory\ \left(t- dt\right)+\left( shipments\ to\ retailer- retail\ sale\right)\times dt $$
$$ initial\ retailer\ inventory=0 $$


$$ shipments\ to\ retailer=\frac{\left( IF\left( wholesaler\ inventory- retailer\ orders\ backlog\ge 0\right) THEN\left( retailer\ orders\ backlog\right) ELSE\left( wholesaler\ inventory\right)\right)}{delivery\ time} $$


$$ retail\ sale= IF\left(\left(\frac{retail er\ inventory}{retail\ delivery\ time}\right)\ge 0\right) THEN(demand) ELSE\left(\frac{retail er\ inventory}{retail er\ delivery\ time}\right) $$
  1. 15.

    Serviceable inventory

    $$ serviceable\ inventory(t)= serviceable\ inventory\ \left(t- dt\right)+\left( production\ rate+ reman\ rate+ acce\ prod\ for\ reuse- shipments\ to\ distributor\right)\times dt $$
$$ initial\ serviceable\ inventory=0 $$


$$ production\ rate=\frac{comp\ used\ for\ prdn}{comp onents\ per\ product} $$
$$ reman\ rate= IF\left(\left(\frac{prod\ for\ reman+ reman\ backlogs}{reprocess\ time}\right)\ge reman\ capacity\right) THEN\left( reman\ capacity\right) ELSE(0) $$
$$ acce\ rate\ for\ reuse=\frac{collected\ products\times reuse\%}{reuse\ inspe\ time} $$


$$ shipment s\ to\ distributor=\frac{\left(\left(\left( serviceable\ inventory- distributor\ orders\ backlog\right)\ge 0\right) THEN\left( distributor\ orders\ backlog\right) ELSE\left( serviceable\ inventory\right)\right)}{distributor\ shipment\ time} $$
  1. 16.

    Wholesaler inventory

    $$ wholesaler\ inventory(t)= wholesaler\ inventory\ \left(t- dt\right)+\left( shipments\ to\ wholesaler- shipments\ to\ retailer\right)\times dt $$
$$ initial\ wholesaler\ inventory=0 $$


$$ shipment s\ to\ wholesaler=\frac{IF\left(\left( distributor\ inventory- wholesaler\ orders\ backlog\right)\ge 0\right) THEN\left( wholesaler\ orders\ backlog\right) ELSE\left( distributor\ inventory\right)}{wholesaler\ shipment\ time} $$


$$ shipments\ to\ retailer=\frac{IF\left(\left( wholesaler\ inventory- retailer\ orders\ backlog\right)\ge 0\right) THEN\left( retailer\ orders\ backlog\right) ELSE\left( wholesaler\ inventory\right)}{delivery\ time} $$
  1. 17.

    Wholesaler orders backlog

    $$ wholesaler\ orders\ backlog\ (t)= wholesaler\ orders\ backlog\ \left(t- dt\right)+\left( wholesaler\ orders- wholesaler\ backlog\ redu\ rate\right)\times dt $$
$$ initial\ wholesaler\ orders\ backlog=0 $$


$$ wholesaler\ orders= expected\ retailer\ orders+\left(\frac{WI\ discrepancy}{WI\ adj\ time}\right) $$


$$ wholesaler\ backlog\ redu\ rate= shipments\ to\ wholesaler $$


$$ aDI=2\times DI\ adjustment\ time $$
$$ aRI=2\times RI\ adjustment\ time $$
$$ as\ usual\ demand= LogNormal\left(4000,2000\right) $$
$$ available\ used\ products= DELAY\left( retail\ sale, residence\ time\right) $$
$$ aWI=2\times WI\ adjustment\ time $$
$$ CI\ adjustment\ time=2 $$
$$ CI\ cover\ time=1.5 $$
$$ CI\ discrepency=\mathit{\operatorname{MAX}}\left(\left( desired\ CI- components\ inventory\right),0\right) $$
$$ compan{y}^{\prime } scollection\ capacity=6000 $$
$$ components\ per\ product=15 $$
$$ component\ production\ capacity=\left(4000\times 15\right) $$
$$ component\ production\ time=\frac{1}{\left(4000\times 15\right)} $$
$$ component\ remanufacturing\ capacity=\left(6000\times 15\right) $$
$$ comp\ replace\%=20 $$
$$ delivery\ time=1 $$
$$ demand= as\ usual\ demand $$
$$ desired\ CI= expected\ distributors\ orders\times components\ per\ product\times CI\ cover\ time $$
$$ desired\ DI= expected\ wholesaler\ orders\times DI\ cover\ time $$
$$ desired\ RI= expected\ demand\times RI\ cover\ time $$
$$ desired\ SI= expected\ mdistributors\ orders\times SI\ cover\ time $$
$$ desired\ WI= expected\ retailer\ orders\times WI\ cover\ time $$
$$ direct\ reuse\%=0 $$
$$ disposal\%=0.01 $$
$$ distributor\ shipment\ time=1 $$
$$ DI\ cover\ time=1.5 $$
$$ DI\ discrepancy=\mathit{\operatorname{MAX}}\left(\left( desired\ DI- distributor\ inventory\right),0\right) $$
$$ DI\ adjustment\ time=2 $$
$$ expected\ demand= SMTH1\left( demand,1\right) $$
$$ expected\ distributor\ orders= SMTH1\left( distributor\ orders, aDI\right) $$
$$ expected\ retailer\ orders= SMTH1\left( retailer\ orders, aRI\right) $$
$$ expected\ wholesaler\ orders= SMTH1\left( wholesaler\ orders, aWI\right) $$
$$ expected\ reman\ products= SMTH1\left( reman\ rate,1\right) $$
$$ expected\ reusable\ components= SMTH1\left( compo\ acce\ for\ dir\ reuse+ comp\ reman\ rate,1\right) $$
$$ initial\ inspection\ time=1 $$
$$ inspection\ and\ dissassembly\ time=2 $$
$$ production\ capacity=4000 $$
$$ products\ not\ collected=\mathit{\operatorname{MIN}}\left(\left( available\ used\ products- collection\ capacity\right),0\right) $$
$$ product\ product ion\ time=\frac{1}{4000} $$
$$ remanufacturing\ capacity=6000 $$
$$ remanufacturing\%=0.75 $$
$$ reprocess\ time=1 $$
$$ residence\ time=16 $$
$$ reuse\%=0 $$
$$ reuse\ inspection\ time=0.001 $$
$$ RI\ cover\ time=1.5 $$
$$ RI\ discrepency=\mathit{\operatorname{MAX}}\left(\left( desired\ RI- retailer\ inventory\right),0\right) $$
$$ RIadjustment\ time=2 $$
$$ SI\ cover\ time=1.5 $$
$$ SI\ discrepancy=\mathit{\operatorname{MAX}}\left(\left( desired\ SI- serviceable\ inventory\right),0\right) $$
$$ SI\ adjustment\ time=2 $$
$$ WI\ cover\ time=1.5 $$
$$ WI\ adjustment\ time=2 $$
$$ WI\ discrepancy=\mathit{\operatorname{MAX}}\left(\left( desired\ WI- wholesaler\ inventory\right),0\right) $$
$$ wholesaler\ shipment\ time=1 $$

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Tombido, L., Louw, L., van Eeden, J. et al. A system dynamics model for the impact of capacity limits on the Bullwhip effect (BWE) in a closed-loop system with remanufacturing. Jnl Remanufactur (2021). https://doi.org/10.1007/s13243-021-00100-7

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  • Bullwhip effect
  • Closed-loop supply chains (CLSCs)
  • Capacity limits
  • Reverse logistics