4. The scheduling problem for additive manufacturing (AM)

The AM scheduling is a problem, different from the traditional single machine scheduling, to be solved since this technology started to be a permanent part of the production environment of several companies especially in the field of defense and aerospace. The model presented here is a summary of what was presented in [39, 40]. The question to which the paper wants to answer is always the same of all scheduling problems, that is:

### "What's the schedule that allows to respect due dates with the least production cost?"

The question is the same but the context as explained in the introduction is very different from the traditional one for the motivations are related to the set-up, that is no more present in the traditional form since it is done all through the design phase and the transferring of the data from the design workstation to the machine and since the fact that with AM it is possible to produce several kinds of geometries in the same production run. For this reason, let us introduce a multi-objective model for the AM schedule that is able to consider also the new constraints given by the new context.

The development framework (Figure 3) is like the ones for the traditional problems of production scheduling, so the production orders are the inputs of AM machine scheduling problem and each order is characterized by the following attributes:


Figure 3. Mathematical model frame.

After that the attributes for the production orders are listed, it is worth to note that in this paper a time and cost model will be applied, in particular, they will be considered the Completion Time (CT) and the Total Part Cost (TPC). CT is the time to produce a single unit of G\_i � th geometry, while TPC is the costs to be covered to produce a single part.

Once the main description elements of our model is described let us to introduce the mathematical formulation of the optimization problem here analyzed. The basic model is taken from a research paper, that used earliness and tardiness as objective function [41], to these objectives in this proposal it is added the cost:


Where:

Substituting the single elements of the (1) with the equations from (2)–(7), it will be possible to

The AM scheduling is a problem, different from the traditional single machine scheduling, to be solved since this technology started to be a permanent part of the production environment of several companies especially in the field of defense and aerospace. The model presented here is a summary of what was presented in [39, 40]. The question to which the paper wants to

The question is the same but the context as explained in the introduction is very different from the traditional one for the motivations are related to the set-up, that is no more present in the traditional form since it is done all through the design phase and the transferring of the data from the design workstation to the machine and since the fact that with AM it is possible to produce several kinds of geometries in the same production run. For this reason, let us introduce a multi-objective model for the AM schedule that is able to consider also the new

The development framework (Figure 3) is like the ones for the traditional problems of production scheduling, so the production orders are the inputs of AM machine scheduling problem and

di: demand of Gi � th geometry or PN ½ � part ddi: due date of Gi � th geometry or PN ½ � day Vi: volume of Gi � th geometry or PN cm<sup>3</sup>

4. The scheduling problem for additive manufacturing (AM)

"What's the schedule that allows to respect due dates with the least production cost?"

answer is always the same of all scheduling problems, that is:

consider the total cost formulation.

82 3D Printing

constraints given by the new context.

Figure 3. Mathematical model frame.

each order is characterized by the following attributes:


The proposed scheduling model has some hypothesis that is listed below:


• The build chamber allows construction of parts on top of each other by support structures or other solutions.

Moreover, as argued in the previous paragraph, the management theme seems to be affected from a very important absence. Many authors have begun to study the management issues related to the general systems; however, nobody recognizes a main limit in the actual knowl-

Therefore, in this chapter, it was presented a cost allocation model that fits the requirements of this new technology and also a mathematical formulation for the scheduling problem of a

1 Department of Engineering, University of Campania "Luigi Vanvitelli", Aversa, CE, Italy

[1] Cima M et al. Three-dimensional printing: The physics and implications of additive

[2] LaMonica M. 2013. https://www.technologyreview.com/s/513716/additive-manufacturing/

[3] Duerden D. 2011. http://www.3d-printing-additive-manufacturing.com/media/down-

[4] Ratnadeep P, Sam A. Optimal part orientation in rapid manufacturing process for achiev-

[5] Arni R, Gupta SK. Manufacturability analysis of flatness tolerances in solid freeform

[6] Hanumaiah N, Ravi B. Rapid tooling form accuracy estimation using region elimination adaptive search based sampling technique. Rapid Prototyping. 2007;13(1):182-190

[7] Ollison T, Berisso K. Three-dimensional printing build variables that impact cylindricity.

[8] Lynn-Charney C, Rosen DW. Usage of accuracy models in stereolithography process

ing geometric tolerances. Journal of Manufacturing Systems. 2011;30:214-222

3 Department of Industrial Engineering, University of Salerno, Fisciano, SA, Italy

, Fabio Fruggiero<sup>2</sup> and Alfredo Lambiase<sup>3</sup>

Production Management Fundamentals for Additive Manufacturing

http://dx.doi.org/10.5772/intechopen.78680

85

edge level.

Author details

Marcello Fera<sup>1</sup>

References

[Online]

single AM machine is presented.

\*, Roberto Macchiaroli<sup>1</sup>

\*Address all correspondence to: marcello.fera@unina2.it

manufacturing. Annals CIRP. 1993;42(1):257-260

Journal of Industrial Technology. 2010;26(1):2-10

planning. Rapid Prototyping Journal. 2000;6(2):77-86

loads/54-d1\_1220\_b-david-duerden\_mbda.pdf. [Online]

fabrication. Journal of Mechanical Design. 1999;123(1):148-156

2 School of Engineering, University of Basilicata, Potenza, PZ, Italy

• Stock costs are neglected.

To solve this kind of problems that are generally reported as NP-hard problems, it is possible to apply several kinds of heuristics such as Tabu Search, genetic algorithm, simulated annealing, ant colonies, bees, particle swarm optimization, and so on.
