Where:

Tsetup: Time required for machine setup [h].

Tmat:ch: Time required to change material [h].

Ni: Quantity of part with i-th geometry [�].

depreciation period and its uptime per year:

Cmachine: Machine cost per hour [€/h].

upt: Machine uptime [hours/year].

3.5. Cost for building up a part

Machine cost: Machine purchase cost [€]. h: Machine depreciation period [years].

all of the parts in the chamber. The cost's items are:

V: Volume of the geometry [cm<sup>3</sup>

Where:

80 3D Printing

• Machine • Energy • Material

• Gas

Fmat:ch: Factor to model the frequency of material changes [�].

].

Finertgas: Factor to model extra effort required for handling in protective gas environment [�].

In the previous formula, it is also possible to include the possibility to work with an extra time of processing due to the use of protective gas (Finertgas). Its value can either be 1 or 0. Also, the change of material can be considered using a 0/1 variable named (Fmat:ch). Furthermore, if the

Machine cost per hour is obtained by dividing the machine purchase cost by the machine

Cmachine <sup>¼</sup> Machine cost

After the presentation of the previous parts of the total production cost, let us to introduce the formula for the calculation of the building step. In this phase, the machine concurrently builds

Therefore, it is possible to define the (6), that is the building cost formulation, which also

<sup>þ</sup> M Gð Þ<sup>i</sup> <sup>∗</sup> Cmaterial∗Wf

includes a waste factor for the powder used in the deposition and sintering phase:

Cbuildð Þ¼ Gi Tbuildð Þ Gi ∗ Cmach þ Cinertgas∗Gascons þ Cenergy∗Pcons∗Ku

<sup>h</sup>∗upt (5)

(6)

costs have to be divided on more build jobs, a fraction can be used in the formulation.

Cbuild: Cost for building up a part with ith geometry [€/part].

Gi: i-th geometry [�].

Tbuild: Total building time [h].

Cmach: Machine cost per hour [€/h].

Cinertgas: Cost of inert gas [€/m<sup>3</sup> ].

Gascons: Average gas consumption [m<sup>3</sup> /h].

Cenergy: Mean energy cost [€/kWh].

Pcons: Power consumption [kW].

Ku: Utilization factor [�].

M: Mass of the geometry [kg].

Cmaterial: Material costs [€/kg].

Wf : Waste factor for powder [�].

#### 3.6. Cost for removing a part from the machine

When the operations of building up are concluded, it is necessary to remove the objects and the substrate plate from the machine chamber. Also, in this case, we included a factor to model, that is, capable to consider the extra time effort for handling in a protective gas environment the production phase. The allocation criteria of this cost are based on parts volume. The formula for this addendum of (1) is reported in (7):

$$\mathbf{C}\_{\text{removal}}(\mathbf{G}\_i) = T\_{\text{removal}} \ast \left( \mathbf{C}\_{\text{op.mach}} + \mathbf{C}\_{\text{mach}} \right) \ast \frac{V(\mathbf{G}\_i)}{\sum\_i V(\mathbf{G}\_i) \ast \mathbf{N}\_i} \ast F\_{\text{imertgas}} \tag{7}$$

Where:

Cremoval: Cost for removing the part with ith geometry from the machine chamber [€/part].

Gi: ith geometry [�].

Tremoval: Time required for removing parts from the machine chamber [h].

Cop:mach: Machine operator's hourly rate [€/h].

Cmach: Machine cost per hour [€/h].

V: Volume of the geometry [cm<sup>3</sup> ].

Ni: Quantity of part with ith geometry [�].

Finertgas: Factor to model extra effort required for handling in protective gas environment [�].

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

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

Production Management Fundamentals for Additive Manufacturing

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

83

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

α<sup>S</sup> : Earliness constant weight ½ � 1=day β<sup>S</sup> : Tardiness constant weight ½ � 1=day Ei : Earliness of i–th geometry ½ � day Ti : Tardiness of i–th geometry ½ � day TOCi: Total Order Cost of the i-th geometry [€] γ<sup>S</sup> : Cost constant weight [1/€] ng: number of order/geometries ½ � � ni,j Number of the i–th item in j–th build ½ � part Vi Volume of i–th geometry cm<sup>3</sup> � � Vchamber Build chamber volume cm<sup>3</sup> � � nb Number of build in the schedule ½ � � di: demand of Gi � th geometry ½ � part

• The scheduling problem faced here is a Single-machine scheduling problem, where the

• The part orientation is given and there is enough space to manually remove the part.

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

G\_i � th geometry, while TPC is the costs to be covered to produce a single part.

FS¼FETþFCP Min!

in this proposal it is added the cost:

<sup>i</sup>¼<sup>1</sup> <sup>α</sup>SEi <sup>þ</sup> <sup>β</sup>STi � �

<sup>i</sup>¼<sup>1</sup> ni,j∗Vi <sup>≤</sup> Vchamber <sup>∀</sup>j∈½ � <sup>1</sup>; nb

αS, βS, γS, TOCi, Vi, Vavailable ∈ R<sup>þ</sup> Ei, Ti, i, j, ng, nb ∈Z<sup>þ</sup>

� �

machine is an AM machine.

<sup>i</sup>¼<sup>1</sup> <sup>γ</sup>STOCi

ni,j ¼ di, ∀i∈ 1; ng

FET <sup>¼</sup> <sup>P</sup>ng

FCP <sup>¼</sup> <sup>P</sup>ng

Subject to: Png

Where:

Pnb j¼1
