## 3 Multi-Objective Optimization Model
### Model Notation
#### 3.3.2 Parameters  
Part-related parameters  
\(c_{i}^{part}\): Printing cost of a part \(i\in I\) ($/part)  
\(c_{i}^{holding}\): Holding cost of a remaining part \(i\in I\) ($/part)  
\(I_{i}\): Projection length of a part \(i\in I\) (millimeter/part)  
\(w_{i}\): Projection width of a part \(i\in I\) (millimeter/part)  
\(h_{i}\): Projection height of a part \(i\in I\) (millimeter/part)  
\(a_{i}\): Projection area of a part \(i\in I\) (millimeter\({}^{2}\)/part)  
\(v_{i}^{convex\_hull}\): Volume of the bounding box of a part \(i\in I\) (millimeter\({}^{3}\)/part)  
\(f^{num\_part}\): Minimum required number of parts to be assigned to operate a 3D printer  
Printer-related parameters  
\(c_{j}^{printer}\): Printer cost for a printer \(j\in J\) ($/printer)  
\(I_{j}^{cap}\): Build chamber length of a printer \(j\in J\) (millimeter/printer)  
\(w_{j}^{cap}\): Build chamber width of a printer \(j\in J\) (millimeter/printer)  
\(h_{j}^{cap}\): Build chamber height of a printer \(j\in J\) (millimeter/printer)  
\(a_{j}^{cap\_max}\): Area capacity for a printer \(j\in J^{ME}\); \(j^{SLA}\) (millimeter\({}^{2}\)/printer)  
\(v_{j}^{cap\_max}\): Volume capacity for a printer \(j\in J^{SLS}\) (millimeter\({}^{3}\)/printer)  
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\(u_{k}^{lim}\): Volume limit of the shipping capacity to each destination \(k\in K\) (millimeter\({}^{3}\))  
\(b^{trans}\): Total planned transportation budget ($)  
Time-related parameters  
\(p_{i}^{part\_ini\_hour}\): Initial print time expected for an individual part \(i\in I\) (hour)  
\(t_{i}^{late}\): Lateness computed from a deviation from the due date for a part \(i\in I\) (day)  
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Here, the binary variable \(U\) and an arbitrarily large number \(m\) are used to restrict a part that exceeds the size of the printer's build chamber. \[l_{i}U_{i,j}^{l}\leq f_{j}^{exp};\forall i\in I,j\in J \tag{16}\]  
\[X_{i,j}\leq U_{i,j}^{l}m;\forall i\in I,j\in J \tag{17}\]  
\[w_{i}U_{i,j}^{w}\leq w_{j}^{exp};\forall i\in I,j\in J \tag{18}\]  
\[X_{i,j}\leq U_{i,j}^{w}m;\forall i\in I,j\in J \tag{19}\]  
\[h_{i}U_{i,j}^{h}\leq h_{j}^{exp};\forall i\in I,j\in J \tag{20}\]  
\[X_{i,j}\leq U_{i,j}^{h}m;\forall i\in I,j\in J \tag{21}\]  
The restriction for the planned production budget, inclusive of the cost of the printer, part printing, and related holding costs, is illustrated in (22). \[\left(\sum_{j\in I}c_{j}^{printer}Y_{j}\right)+\left(\sum_{i\in I}\sum_{j\in I} c_{i}^{part}X_{i,j}\right)+\left(\sum_{i\in I}c_{i}^{holding}R_{i}^{prod}\right)\leq b ^{prod} \tag{22}\]  
As the sizes and geometries of parts are varied and must be printed layer-by-layer, the print time varies by part and by AM technology. The ME technology uses a heated extruder to dispense a filament material from an extruder layer-by-layer and part-by-part; thus, the total print time for an ME printer can be computed based on the sum of the print times of all the assigned parts, as shown in (23). Conversely, SLS and SLA use laser technology, which scans each layer of all scheduled parts such that the powdered material in SLS is sintered and the photopolymer resin in SLA is cured approximately at the same time. As a result, the print time can be computed using the maximum print time based on all parts assigned to the printer since the assigned part with the greatest time requirement will determine the print time (24). This non-linear term was linearly transformed using (25)-(27), in which the auxiliary variable \(S_{i,j}\) is used in the transformation process. Next, the print time in days is computed based on the print time in hours and labor working time (28).