If the­re are mul­ti­ple dif­fe­rent resour­ces, pro­ces­sing machi­nes, and cons­traints pre­sent, or if a dif­fe­rent qua­li­ty cri­ter­ion is to be opti­mi­zed, then the­re likely exist no opti­mal sche­du­les that can be manu­al­ly con­s­truc­ted fol­lo­wing simp­le rules.

Ins­tead, optimi­zation algo­rith­ms must be employ­ed. Often, the­se are mixed-inte­ger optimi­zation pro­blems, which are com­pu­ta­tio­nal­ly chal­len­ging to sol­ve. A sto­rage cost-mini­mi­zing sche­du­le for \(n\) jobs on \(m\) machi­nes can be for­mu­la­ted as the optimi­zation pro­blem [1, p. 134]

$$\begin{align} \min ~~~& \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^nR_{ijk} p_{ij} x_{ikj} && \\ \text{s.t.} ~~~& \sum_{i=1}^m \sum_{k=1}^n x_{ikj}=1 &&j=1, …, n  \\ & \sum_{j=1}^m x_{ikj} \le 1 && i=1, …,m ~ k=1, …, n \\ & x_{ikj}\in \{0,1\} && i=1, …, m ~ k=1, …, n ~ j=1, …, n \end{align}$$

Here, \(p_{ij}\) is the pro­ces­sing dura­ti­on of job \(j\) on machi­ne \(i\), and \(x_{ikj}\) is a bina­ry indi­ca­tor varia­ble signi­fy­ing whe­ther job \(j\) is exe­cu­ted on machi­ne \(i\) in the \(k\)-th last posi­ti­on. The cons­traints ensu­re that each job is assi­gned exact­ly once. The term \(R_{ijk}\) repres­ents the resour­ce quan­ti­ties that need to be stored at the \(k\)-th last moment for pro­ces­sing on machi­ne \(i\). The­se are the resour­ce quan­ti­ties for later pro­jects on machi­ne \(i\) that need to be stored over the dura­ti­on \(p_{ij}\). The­r­e­fo­re, it follows:

$$R_{ijk} = \begin{cases} kr ~~~~~~~~~~~\text{ if } r_1=r_2= … = r_n =:r \\ \sum_{l\le k} x_{ikj}r_j \text{ other­wi­se }\end{cases} $$

In the case of une­qual resour­ce con­sump­ti­on, the optimi­zation pro­blem is non­line­ar in the indi­ca­tor varia­bles \(x_{ikj}\) and can only be sol­ved appro­xi­m­ate­ly, e.g., through rela­xa­ti­ons based on semi­de­fi­ni­te pro­gramming [2].