3. The production cost model

The present model is summarized and reported by another paper of the authors appeared on 2017 [35–37].

#### 3.1. Total manufacturing cost

The total manufacturing cost for each geometry is obtained by summing the cost of each step to finish a part belonging to a specific client order. The single elements of this formula will be developed and illustrated in the following part of this section, giving the possibility to anyone wants to replicate the cost calculation scheme here reported.

$$\mathbf{C}\_{\text{tot}}(\mathbf{G}\_{i}) = \mathbf{C}\_{\text{prep}}(\mathbf{G}\_{i}) + \mathbf{C}\_{\text{build}]}(\mathbf{G}\_{i}) + \mathbf{C}\_{\text{setup}}(\mathbf{G}\_{i}) + \mathbf{C}\_{\text{build}}(\mathbf{G}\_{i}) + \mathbf{C}\_{\text{removed}}(\mathbf{G}\_{i}) \tag{1}$$

Tprep: Time required for preparing CAD data [h].

Ni; Quantity of the part with ith geometry [�].

Cbuildjob: Cost for build job assembly [€/part].

Tbuildjob: Time required for build job assembly [h].

Cop:pre: Pre-processing operator's hourly rate [€/h].

and the final formulation is as the one reported in (4):

].

geometries inserted in the build camera, as reported in (3):

In the traditional approach, as reported in Rickenbacher et al. [38] this cost is allocated equally between all parts while in the present formulation this cost is allocated with a ratio that considers the how much volume each geometry occupies in the total volume given by the

> V Gð Þ<sup>i</sup> <sup>i</sup> V Gð Þ<sup>i</sup> ∗Ni

Production Management Fundamentals for Additive Manufacturing

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

V Gð Þ<sup>i</sup> <sup>i</sup> V Gð Þ<sup>i</sup> ∗Ni

(4)

(3)

79

Cbuildjobð Þ¼ Gi Tbuildjob<sup>∗</sup> Cop:pre <sup>þ</sup> CPC � �∗<sup>P</sup>

CPC: Hourly rate of the workstation including costs of required software and tools [€/h].

When all the previous activities, that is, the preparing geometry and the planning phase are completed the real production phase can start. This phase includes the data import and machine setup phases. During this time, the machine cannot be used, and for this reason, we included its hourly cost. Also, in this case, we used the parts volume like the allocation criteria

Csetupð Þ¼ Gi Cop:mach <sup>þ</sup> Cmach � �<sup>∗</sup> Tsetup <sup>þ</sup> ð Þ Fmat:ch∗Tmat:ch � �∗Finertgas∗<sup>P</sup>

3.3. Cost for building job assembly

Where:

Where:

Gi: i-th geometry [�].

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

Csetup: Machine setup costs [€/part].

Cmach: Machine cost per hour [€/h].

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

Gi: i-th geometry [�].

3.4. Machine setup costs

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

Where:

Ctot: Total manufacturing cost of each part with ith geometry [€/part].

Gi: ith geometry [�].

Cprep: Cost for preparing geometry data (orientation, support structures, etc.) [€/part].

Cbuildjob: Cost for build job assembly [€/part].

Csetup: Machine setup costs [€/part].

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

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

#### 3.2. Cost for preparing geometry data

The first step of the process reported here is the design of the geometry data that includes orientation and support structure generation for each geometry. A possible formulation for this cost considers the specific value of a cost referred to the number of parts to be produced for each geometry, as reported in (2):

$$\mathbf{C}\_{prop}(\mathbf{G}\_i) = \left(\mathbf{C}\_{op, pre} + \mathbf{C}\_{PC}\right) \* \frac{T\_{prop}(\mathbf{G}\_i)}{N\_i} \tag{2}$$

Where:

Cprep: Cost for preparing geometry data (orientation, support structures, etc.) [€/part].

Gi: ith geometry [�].

Cop:pre: Pre-processing operator's hourly rate [€/h].

CPC: Hourly rate of the workstation including costs of required software and tools [€/h].

Tprep: Time required for preparing CAD data [h].

Ni; Quantity of the part with ith geometry [�].

#### 3.3. Cost for building job assembly

In the traditional approach, as reported in Rickenbacher et al. [38] this cost is allocated equally between all parts while in the present formulation this cost is allocated with a ratio that considers the how much volume each geometry occupies in the total volume given by the geometries inserted in the build camera, as reported in (3):

$$\mathbb{C}\_{bubble}(G\_i) = T\_{buildjvb} \* \left(\mathbb{C}\_{op, pre} + \mathbb{C}\_{P\mathbb{C}}\right) \* \frac{V(G\_i)}{\sum\_i V(G\_i) \* N\_i} \tag{3}$$

Where:

3. The production cost model

3.1. Total manufacturing cost

wants to replicate the cost calculation scheme here reported.

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

Ctot: Total manufacturing cost of each part with ith geometry [€/part].

Cprep: Cost for preparing geometry data (orientation, support structures, etc.) [€/part].

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

The first step of the process reported here is the design of the geometry data that includes orientation and support structure generation for each geometry. A possible formulation for this cost considers the specific value of a cost referred to the number of parts to be produced for

∗

Cprep: Cost for preparing geometry data (orientation, support structures, etc.) [€/part].

CPC: Hourly rate of the workstation including costs of required software and tools [€/h].

Tprepð Þ Gi Ni

(2)

Cprepð Þ¼ Gi Cop:pre þ CPC

2017 [35–37].

78 3D Printing

Where:

Where:

Gi: ith geometry [�].

Gi: ith geometry [�].

Cbuildjob: Cost for build job assembly [€/part].

Csetup: Machine setup costs [€/part].

3.2. Cost for preparing geometry data

each geometry, as reported in (2):

Cop:pre: Pre-processing operator's hourly rate [€/h].

The present model is summarized and reported by another paper of the authors appeared on

The total manufacturing cost for each geometry is obtained by summing the cost of each step to finish a part belonging to a specific client order. The single elements of this formula will be developed and illustrated in the following part of this section, giving the possibility to anyone

Ctotð Þ¼ Gi Cprepð Þþ Gi Cbuildjobð Þþ Gi Csetupð Þþ Gi Cbuildð Þþ Gi Cremovalð Þ Gi (1)

Cbuildjob: Cost for build job assembly [€/part].

Gi: i-th geometry [�].

Tbuildjob: Time required for build job assembly [h].

Cop:pre: Pre-processing operator's hourly rate [€/h].

CPC: Hourly rate of the workstation including costs of required software and tools [€/h].


#### 3.4. Machine setup costs

When all the previous activities, that is, the preparing geometry and the planning phase are completed the real production phase can start. This phase includes the data import and machine setup phases. During this time, the machine cannot be used, and for this reason, we included its hourly cost. Also, in this case, we used the parts volume like the allocation criteria and the final formulation is as the one reported in (4):

$$\mathbf{C\_{setup}}(\mathbf{G\_{i}}) = \left(\mathbf{C\_{op.mach}} + \mathbf{C\_{mach}}\right) \ast \left(T\_{setup} + \left(F\_{\text{mut.ch}} \ast T\_{\text{mut.ch}}\right)\right) \ast F\_{\text{interfages}} \ast \frac{V(\mathbf{G\_{i}})}{\sum\_{i} V(\mathbf{G\_{i}}) \ast N\_{i}} \tag{4}$$

Where:

Csetup: Machine setup costs [€/part].

Gi: i-th geometry [�].

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

Cmach: Machine cost per hour [€/h].

Tsetup: Time required for machine setup [h].

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

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

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

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

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

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

Machine cost per hour is obtained by dividing the machine purchase cost by the machine depreciation period and its uptime per year:

$$\mathcal{C}\_{machine} = \frac{\text{Machine cost}}{\text{h} \ast \text{upt}} \tag{5}$$

Where:

Where:

Gi: ith geometry [�].

Gi: i-th geometry [�].

Tbuild: Total building time [h].

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

Cmach: Machine cost per hour [€/h].

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

Cenergy: Mean energy cost [€/kWh].

Pcons: Power consumption [kW].

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

formula for this addendum of (1) is reported in (7):

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

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

Cmach: Machine cost per hour [€/h]. V: Volume of the geometry [cm<sup>3</sup>

Ku: Utilization factor [�].

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

].

/h].

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

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

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

V Gð Þ<sup>i</sup> <sup>i</sup> V Gð Þ<sup>i</sup> ∗Ni

Production Management Fundamentals for Additive Manufacturing

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

81

∗Finertgas (7)

Cremovalð Þ¼ Gi Tremoval<sup>∗</sup> Cop:mach <sup>þ</sup> Cmach � �∗<sup>P</sup>

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

].

Where:

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

Machine cost: Machine purchase cost [€].

h: Machine depreciation period [years].

upt: Machine uptime [hours/year].

#### 3.5. Cost for building up a part

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


Therefore, it is possible to define the (6), that is the building cost formulation, which also includes a waste factor for the powder used in the deposition and sintering phase:

$$\mathbb{C}\_{\text{build}}(\mathbf{G}\_{i}) = T\_{\text{build}}(\mathbf{G}\_{i}) \ast \left( \mathbb{C}\_{\text{much}} + \mathbb{C}\_{\text{interly}} \ast \mathbb{G} \mathbf{a} \mathbf{s}\_{\text{cons}} + \mathbb{C}\_{\text{energy}} \ast \mathbb{P}\_{\text{cons}} \ast \mathbf{K}\_{\text{u}} \right) + M(\mathbf{G}\_{i}) \ast \left( \mathbb{C}\_{\text{multi-ál}} \ast \mathbb{W}\_{f} \right) \tag{6}$$
