# Linking Scheduling Criteria to Shop Floor Performance in Permutation Flowships
## 2 Background and Related Work
### Shop Floor Performance Measures  
Shop floor performance is usually measured using different indicators. Among classical texts, [PERSON] [33] mentions throughput, inventory, and operating expenses as key manufacturing performance measures. [PERSON] [34] mentions the following manufacturing objectives: meet due dates, minimise WIP, minimise cycle time, and achieve a high resource utilisation. [PERSON] [35] identifies four main objectives in the production process: short lead times, low schedule deviation, low inventories, and high utilisation. [PERSON] and [PERSON] [1] list the following manufacturing objectives: high throughput, low inventory, high utilisation, short cycle times, and high product variety. [PERSON] et al. [36] cites utilisation and work-in-process as the two main managerial concerns in manufacturing systems. Finally, throughput and lateness are identified by several authors (e.g., [37; 38]) as the main performance indicators in manufacturing. Although these objectives have remained the same during decades [39], their relative importance has changed across time [40], and also depends on the specific manufacturing sector (for instance, in the semiconductor industry, average cycle time is regarded as the most important objective, see, e.g., [41] or [42]). According to the references reviewed above, we consider three performance measures: Throughput (\(TH\)), Work-In-Process (\(WIP\)), and Average Cycle Time (\(ACT\)) as shop floor performance indicators. With respect to other indicators mentioned in the reviewed references, note that one of them is not relevant in the deterministic environment to which this analysis is constrained (low schedule deviation), while other is not specifically related to shop floor operation (high product variety). Furthermore, as our study does not assume individual due dates for jobs, we exclude due date related measures, although we wish to note that, quite often short cycle times are employed as an  
Figure 1: Different components of machine idle time. indicator of due date adherence [38; 43]. Finally, we prove below that utilisation and throughput are directly related, so utilisation does not need to be considered in addition to throughput. Regarding the relationship of the shop floor performance measures with the scheduling criteria, it is easy to check that \(TH\) the throughput may be defined in terms of \(C_{max}(S)\) the makespan of a sequence \(S\) of \(n\) jobs, i.e.,  
\[TH(S)=\frac{n}{C_{max}(S)} \tag{1}\]  
As a result, throughput is inversely proportional to makespan. Note that the utilisation \(U(S)\) can be defined as (see, e.g., [36]):  
\[U(S)=\frac{\sum_{i}\sum_{j}p_{ij}}{C_{max}(S)} \tag{2}\]  
therefore, it is clear that \(U(S)=\frac{\sum_{i}\sum_{j}p_{ij}}{n}\cdot TH(S)\), and, as \(\frac{\sum_{i}\sum_{j}p_{ij}}{n}\) is constant for a given instance, then it can bee seen that the two indicators are fully related. Accordingly, \(ACT\) average cycle time can be expressed in terms of the completion time, see, e.g., [44]:  
\[ACT(S)=\frac{\sum C_{j}(S)}{n} \tag{3}\]  
It follows that the total completion time is proportional to \(ACT\). Since \(TH\), \(ACT\) and \(WIP\) are linked through [PERSON]'s law, the following equation holds. \[WIP(S)=TH(S)\cdot ACT(S)=\frac{\sum C_{j}(S)}{C_{max}(S)} \tag{4}\]  
From Equation (4), it may be seen that total completion time and \(WIP\) minimisation are not exactly equivalent, although it is a common statement in the scheduling flowshop literature: It is easy to show that the two criteria are equivalent for the single-machine case, but this does not necessarily hold for the flowshop case. As, apart from the two theoretical equivalences above discussed, there are no straightforward relationship between the scheduling criteria and the shop floor performance measures, such relationships should be empirically discovered over a high number of problem instances. This computational experience must take into account that the results might be possibly influenced by the instance sizes and the processing times employed.