# Push/Pull Parts Production Policy Optimization in the ATO Environment
## 3 Bi-Objective Model for Push/Pull Policy Selection in an ATO Environment
### Objective Functions  
The two basic objective functions are:  
\(P_{TOT}\)\(T_{ATOT}\)  
environment. The mathematical optimization model aims to minimize storage space and setup times. The procedure is based on four main steps and a schematic flowchart is reported in Figure 2.  
STEP 1. Calculation of the parameters \(a_{i}\) and \(EPQ_{i}\) for each part \(i\). In this way, it is possible to graphically define the position of each part. The suitable parts for a Pull policy will be the parts with high values of \(a_{i}\) and low values of \(EPQ_{i}\) (as illustrated in Figure 3). In fact, to produce a part \(i\) with a Pull approach, it would be necessary to obtain high values of \(a_{i}\), representing the capacity of the upstream production system to rapidly change the parts to produce and to quickly produce a low number of items instead of an entire production batch. As a consequence, for a part \(i\), low setup time \(T_{STOT_{i}}\) Equation (2) and low production time \(T_{CTOT_{i}}\) Equation (3) are required within the whole production process (high value of agility \(a_{i}\)) to apply a proper Pull policy. Low values of \(EPQ_{i}\), meaning that the most critical parts for a Push policy are those with high dimensions or critical shapes. Consequently, for part \(i\), low values of \(EPQ_{i}\) are required to apply a proper Pull policy. Figure 3 reports the potential Push-Pull zones in the \(a_{i}\) and \(EPQ_{i}\) diagram. The optimisation problem lies on the optimal definition of the Pull zone, i.e., the breakeven optimal values of the two parameters a** and EPQ**. STEP 2. Multiscenario analysis. Considering different values of the two parameters (as illustrated in Figure 3), the values of the two objective functions \(P_{TOT}\) and \(TA_{TOT}\) are assessed. Main results are in Figure 4 (blue dots) where for each scenario the two objective functions values are plotted. Figure 2: Algorithm for push/pull policy selection. The procedure adopted to generate and assess the different scenarios is detailed reported in Figure 2 (Step 2). A parameter, i.e., \(\epsilon\), is introduced that can be chosen small enough to generate a representative number of potential scenarios. STEP 3. Construction of the Pareto frontier of the optimal solutions and breakeven optimal parameters a** and EPQ** definition. According to [PERSON] et al. (2012), starting from the Step 2 results, the dominated and the dominant solutions are found to define the Pareto Frontier. Figure 4 conceptually reports the Pareto Frontier (red curve) of the different scenarios analyzed in the Step 2 with the points. The set of the points belonging to the Pareto Frontier are defined as \(\overline{(TA_{TOT};P_{TOT})}\). At this stage, the decision maker needs to select an optimal trade-off solution \(\overline{(\overline{TA_{TOT}};\overline{P_{TOT}})}\), among the set of dominant points ([PERSON] et al., 2012). According to [PERSON] et al. (2012) the Utopia point is the point in the solution space corresponding to the optimal  
Figure 4: Objective functions plot for different Push/Pull scenarios. Figure 3: Potential Push-Pull zones in \(a_{i}\) and \(EPQ_{i}\) diagram. values for all criteria. In this case, the Utopia point is the point where the total setup time and the total number of stored EPAL are set to 0:  
\[U_{P_{TOT}},\ U_{TA_{TOT}}=(0;0) \tag{10}\]  
To find the bi-objective optimal point, a method based the minimization of the Euclidean distance between the Utopia point and the points belonging to the Pareto frontier is proposed. It can be defined finding the point \((\overline{\overline{TA_{TOT}}};\overline{\overline{P_{TOT}}})\) belonging to the Pareto frontier that minimizes:  
\[\min\sqrt{\left(\left(\overline{TA_{TOT}}-U_{TA_{TOT}}\right)^{2}+\left( \overline{P_{TOT}}-U_{P_{TOT}}\right)^{2}\right)} \tag{11}\]  
STEP 4. Final Push/Pull policy definition for each item. In this step, according to the breakeven optimal values of a** and EPQ**, it is possible to define the optimal Push/Pull policy to adopt for each part \(i\).