## 7 New Settings of Simple Assembly Line Balancing Problems and Unresolved Issues
### Maximization of the Effectiveness of Assembly Line Modifications  
Due to a significant change in the conditions of assembly production, it may be necessary to solve the following optimization problem, which we denote SALBP-\(E_{\alpha}^{\beta}\). Suppose there is a set, \(S=\{S_{1},S_{2},\ldots,S_{m}\}\), of workstations that can be used in assembly production. The duration, \(t_{i},i\in V\), of the assembly operations and the partial strict order of their execution, defined by the digraph \(G=(V,A)\), are also specified. The average costs \(\alpha\geq 0\), for using one workstation from the selected workstation subset of the set, \(S=\{S_{1},S_{2},\ldots,S_{m}\}\), of the workstations available at the enterprise. This cost, \(\alpha\geq 0\), includes depreciation of one workstation, energy costs, creation and exploiting of a new workstation, and commissioning of a previously reserved workstation. The average cost \(\beta\geq 0\) of assembling one product on the assembly conveyor is usually known before solving the problem. In the SALBP-\(E_{\alpha}^{\beta}\), it is necessary to find an optimal line balance, \(b=(V_{1}^{b},V_{2}^{b},\ldots,V_{m}^{b})\), of assembly operations, \(V\), and the cardinality, \(m\), of a subset of workstations used on the assembly conveyor. Using the optimal line balance is required to minimize the following weighted costs:  
\[E_{\alpha}^{\beta}=\alpha\cdot m+\beta\cdot c \tag{2}\]  
for the exploitation of the assembly conveyor and the assembly of the final products of the assembly conveyor. Recall that the SALBP-E proposed in [2] requires maximizing the efficiency of the assembly line, which is determined by the Equality (1). If the relative costs of the exploitation of the assembly line and the production of assembly products are considered and the equality \(\alpha+\beta=1\) is assumed, then the problem SALBP-\(E_{\alpha}^{\beta}\) turns into the problem SALBP-1 with the equality \(\beta=0\), and into the problem SALBP-2 with the equality \(\alpha=0\). If the absolute cost of the exploitation of the assembly line and the production of assembly products are considered, then the objective function of the SALBP-\(E_{\alpha}^{\beta}\) takes the form:  
\[E_{\alpha}^{\beta}=\alpha\cdot\left(\sum_{i=1}^{m}E_{i}+\sum_{j=m+1}^{m+k}E_{j} \right)+\beta\cdot c, \tag{3}\]  
in the case of increasing the set of workstations of the assembly line by including \(k\geq 1\) new workstations in the assembly production. The objective function of the SALBP-\(E_{\alpha}^{\beta}\) is of the form:  
\[E_{\alpha}^{\beta}=\alpha\cdot\left(\sum_{i=1}^{m}E_{i}-\sum_{j=k}^{m}E_{j} \right)+\beta\cdot c \tag{4}\]  
if it is needed to reduce the set of workstations of the existing assembly line by excluding \((m-k+1)\) workstations from the assembly production. In the Formula (4), the value of \(E_{i}\) determines the total cost of the exploitation of a workstation \(S_{i}\) from a set of the workstations, \(S=\{S_{1},S_{2},\ldots,S_{m}\}\), used on an operating assembly line. The Formula (3) defines the values of \(E_{i}\) for all the workstations, \(S_{i}\in S\). For workstations of the set, \(\{S_{m+1},S_{m+1},\ldots,S_{m+k}\}\), the value of \(E_{j},j\in\{m+1,m+2,\ldots,m+k\}\), in addition to the total cost of operating a workstation \(S_{j}\) includes the cost of creating a workstation, if it is a new workstation, and the cost of putting it into operation, if the workstation \(S_{j}\) was previously reserved. If the set of workstations in use is increased to a set, \(S^{+}=\{S_{1},S_{2},\ldots,S_{m},S_{m+1},\ldots,S_{m+k}\}\), it is necessary to renumber all the workstations in use so that when the workstations are indexed again, the inequality, \(u<v\), implies that the workstation \(S_{u}\) precedes the workstation \(S_{v}\) in the modified assembly line, \(S^{+}=\{\ldots,S_{u},\ldots,S_{v}\ldots\}\).