# An Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem
## 3 Problem Description and Formulation
### Mathematical Model
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Description** \\ \hline Index: & \\ \(i\) & Operation. \((i=1,2,\ldots,m)\). \\ \(j\) & Station. \((j=1,2,\ldots,n)\). \\ \((i,r)\) & Resource. \(\left((i,r)=(1,1),\ldots,(1,r_{1}^{\max}),\ldots,(m,r_{m}^{\max})\right)\). \\ \(l\) & Operation index of the station. \((l\in L=\{1,2,\ldots,l_{\max}\})\). \\ \(k\) & Resource index of the station. \((k\in K=\{1,2,\ldots,k_{\max}\})\). \\ Parameter: & \\ \(D\) & Set of dedicated resource. \\ \(R\) & Set of reconfigurable resource. \\ \(F\) & Set of flexible resource. \\ \(AL\) & Set of resources from old assembly line. \\ \(z_{1}\) & Number of resources in old assembly line. \\ \(m\) & Number of operations. \\ \(ct\) & Cycle time. \\ \(\varphi\) & Cost of additional dedicated resource. \\ \(\kappa\) & Cost of additional reconfigurable resource. \\ \(\lambda\) & Cost of additional flexible resource. \\ \(\mu\) & Cost of moving a resource. \((\mu\ll\varphi,\kappa,\lambda)\). \\ \(SR^{\prime}_{jk}\) & Resource \(k\) in station \(j\). \((SR^{\prime}_{jk}=(i,r))\). \\ \(r_{i}^{\max}\) & Maximum number of resources of operation \(i\). \\ \(to_{ir}\) & Process time of resource \(r\) of operation \(i\). \\ \(P_{i}\) & Set of precursor operation for operation \(i\). \\ \(S_{i}\) & Set of successor operation for operation \(i\). \\ \(ts_{j}\) & Process time of station \(j\). \\ \(dom_{i,r}\) & Operation index of resource. \(dom_{i,r}=i\). \\ Decision variable: & \\ \(n\) & Number of stations. \\ \(SO_{jl}\) & Operation \(l\) in station \(j\). \((SO_{jl}=1,2,\ldots,m)\). \\ \(SR_{jk}\) & Resource \(k\) in station \(j\). \((SR_{jk}=(i,r))\). \\ \(SRN_{jk}\) & Whether resource \(k\) in station \(j\) is a new resource. \\ \((SRN_{jk}=0,1)\). & \\ \(SRO_{jk}\) & Whether resource \(k\) in station \(j\) is from old assembly line. \\ \(SRQ_{jk}\) & Number of resource \(k\) in station \(j\). \((SRQ_{jk}\in N)\). \\ \(SRO^{\prime}_{jk}\) & Resource \(k\) in station \(j\) is a move of the resource from other station in old assembly line. \((SRO^{\prime}_{jk}=0,1)\). \\ \(SRD_{jk}\) & Resource \(k\) in station \(j\) is a new dedicated resource. \\ \(SRR_{jk}\) & Resource \(k\) in station \(j\) is a new reconfigurable resource. \\ \(SRF_{jk}\) & Resource \(k\) in station \(j\) is a new flexible resource. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Formula (1) minimizes the reconfiguration cost of the assembly line. This study subdivides the reconfiguration cost into \"cost of moving a station\", \"cost of moving a resource\", \"cost of additional dedicated resource\", \"cost of additional reconfigurable resource\" and \"cost of additional flexible resource\". The cost of moving a station is set to 0. Other costs are assigned as \(\varphi\), \(\kappa\), \(\lambda\), and \(\mu\), respectively. The specific values of these costs can be obtained through on-site surveys of enterprises or determined based on the production preferences of the enterprises. Constraint (2) restricts the resources of new assembly line are from either old assembly line or resource library. Constraint (3) and (4) restrict the decision variable to the values \(\{0,1\}\). Constraints (5)-(7) express all constraints regarding resources from the old assembly line. Constraint (5) restricts the usage of resources from the old assembly line to only once in the new assembly line. Constraint (6) determines whether resources within stations of the old assembly line are relocated as a whole. Constraint (7) restricts the decision variable to the values \(\{0,1\}\). Constraints (8)-(14) express all constraints regarding resources from resource library. Constraint (8) specifies whether a resource is dedicated. Constraint (9) restricts the decision variable to the values \(\{0,1\}\). Constraint (10) specifies whether a resource is reconfigurable. Constraint (11) restricts the decision variable to the values \(\{0,1\}\). Constraint (12) specifies whether a resource is flexible. Constraint (13) restricts the decision variable to the values \(\{0,1\}\).