# Push/Pull Parts Production Policy Optimization in the ATO Environment
## 4 Case Study  
The case study is referred to an Italian washing machine manufacturer. In the case study, the parts category where the proposed model for Push/Pull policy definition was applied are the metal sheet plate parts (as illustrated in Figure 6 and Table 2). They are used as the product structure, generally used in the first assembly phases, and as product covers, which are generally used in the last assembly phases. They are interesting because:  
1. The production technologies used to produce these parts are cutting, bending, and welding. They are common, with not a high level of know-how. 2. Their value can be considerable, especially if produced with high-value steels (i.e., stainless steel). 3. They have a wide production mix (in terms of product variant dimension, shape, colors, etc.) since they are frequently used for the product customization. 4. They are large and voluminous. Figure 5: Final optimal PUSH/PULL policy definition for different parts i. The last two points, considering a pure Push approach, negatively affect the inventory levels, with a potentially large occupation volume within the warehouse. For this reason, many industrial companies decide to include in their internal production the cutting, blending, welding, and if necessary, painting processes. A total of 425 internally produced parts are considered in the analysis, previously produced with an MTS-Push approach. In this way it is possible to graphically define the position of each part according to the two parameters. The suitable parts for a Pull policy will be the parts with high values of \(a_{i}\) and low values of \(EPQ_{i}\). Figure 7 lists the 425 parts characterizing the industrial case study. STEP 2. Multiscenario analysis. Considering different values of a* and EPQ* (as illustrated in Figure 7), the values of the two objective functions \(P_{TOT}\) Equation (6) and \(TA_{TOT}\) Equation (8) are assessed. Main results are in Figure 8, where for each scenario the two objective functions values are plotted. \begin{table}
\begin{tabular}{c c c} \hline _Item_ & _ai_ & _EPQi_ \\ \hline
660,622 & 30.2 & 28 \\ \hline
7,086,197 & 10.3 & 6 \\ \hline
806,614 & 45.5 & 28 \\ \hline
100,635 & 34.2 & 536 \\ \hline
100,645 & 10.4 & 112 \\ \hline
100,708 & 91.3 & 112 \\ \hline
806,615 & 58.4 & 28 \\ \hline
110,616 & 12.9 & 20 \\ \hline
100,622 & 13.7 & 22 \\ \hline \end{tabular}
\end{table} 
Table 2: Case study \(a_{i}\) and \(EPQ_{i}\) values. Figure 6: Example of case study part. The procedure adopted to generate the different scenarios is detailed reported in Figure 2 (Step 2). From the computational point of view, the problem is trivial. The multiscenario analysis reported in Figure 4 was performed using Matlab SW with an Intel(R) Core i7 getting 2.88 \(\times\) 10\({}^{5}\) scenarios in 9.89 s.  
STEP 3. Construction of the Pareto Frontier of the optimal solutions and breakeven optimal parameters a** and EPQ** definition. Figure 9 reports the Pareto Frontier of the case study, the set of solutions where no other solution dominates them. At this stage, the decision maker needs to select an optimal bi-objective solution \(\overline{(TA_{\text{TOT}},\overline{P_{\text{TOT}}})}\) among the set of dominant points. To find the bi-objective optimal point, the minimization of the Euclidean distance between the Utopia point and the points belonging were applied Equation (11). In Figure 9, the bi-objective optimum point (in red) within the Pareto frontier (in blue) compared to that of the Utopia point (in yellow) is reported. Figure 8: Case Study objective functions plot for different Push/Pull scenarios. Figure 7: Case study \(a_{i}\) and \(EPQ_{i}\) calculation and potential MTO zone. The bi-objective optimum point Equation (11) is:  
\((\overline{\left(\overline{TA}_{\text{TOT}}:\overline{\overline{P}_{\text{TOT}}} \right)})=(305.72;185.25)\). Once defined it, it is possible to derive the breakeven optimal values of a** and EPQ** of the related scenario that are a** = 29.75 pieces/h2 and EPQ** = 12 pieces/pallet. STEP 4.