5.2 Process Based Cost Model
5.2.1 Model Development
To develop the process based cost model for this work, it is necessary to divide the final cost in all its individual costs: material, energy, labour, machine, tooling and maintenance. These costs are also divided in variable and fixed costs, where the first are directly dependent on the annual production volume and the latter are computed by allocating annual cost [24].
The choice to divide the final cost in these individual costs was made due to their influence in the final cost, where the most important are material, energy and mould, and as a result of the variables applied thought the study that will influence these parameters. Still, these drivers are not exclusive in the sense that others exist but contribute minutely to the overall cost, or are dependent on exterior factors, as for example overhead costs which are more dependent on the size of the company than influenced by processing parameters. All these values, dependent on the company, are excluded from this study, since the model is created excluding a company basis. All remaining aspects will affect the overall part cost in a smaller or greater way, depending on the material and part to be produced.
A final note on this model is that it only considers the design and production phased of a certain part, since the objective of this work is to create a tool to support decision making in the mould design phase, disregarding the use and end-of-life of the final part. In terms of mould production, the manufacturing processes are not studied, with the processing times defined as a constant, only dependent on the type of feeding system and number of cavities defined.
To find each individual cost, all inputs of part geometry, material, machine, mould and process must be defined, as previously stated, with a number of mould dimensions and machines required, in order for the model to choose the best option within that list. With this, the model calculates all necessary intermediate variables and finally the costs and environmental impact.
To better understand the model developed for this work, each cost is presented individually, having its own flowchart, facilitating the review of how each cost is calculated and what inputs are the most influential, and demonstrating the complexity of the model. Still, as several costs are expected to produce a low impact on the overall cost, the PBCM flowcharts for labour, machine, building and maintenance costs will be presented in Annex C. Since there are several inputs to each cost, but most
35 are included in more than one calculation, a colour code is created dividing the inputs in material, part, mould and process data. This code, used to help identify the origin of each input variable in the following flowcharts for the various costs, is presented in Figure 5.3 and the respective variables for each type of data are presented in Annex A.
Figure 5.3 - Colour Code for Model Data
5.2.1.1 Material Cost
Material cost is a variable cost, meaning it will be a factor of production volume as presented in Equation (5.1), an adaptation of the equation from [24].
Material Cost [€] = (Effective Production Volume [parts] Part Weight [kg/part]+
Setup Scrap per year [kg]+Engineered Scrap per year[kg]) Cost [€/kg] - Effective Recycle per year [kg] Cost[€/kg]
(5.1)
Since this model considers that material may or may not be recycled, as a design choice, and the wasted material is a sum of material from the setup operation and the additional material from the runners and sprue, most of the previous formula is explained. Another important parameter is the reject rate, which in this case enters the equation in the effective production volume. The individual Equations (5.2) through (5.8), for each factor, are presented next, as a means to illustrate all the involved parameters in this cost [24, 32].
Annual Production Volume[part]
Effective Production Volume[part] =
1 - Reject Rate[%] (5.2)
3 3
Part Weight[kg/part] = [kg/m ] Part Volume[m /part]
(5.3) Scrap per Setup[kg]×Effective Production Volume[part]Setup Scrap per year[kg] =
Batch[part] (5.4)
Engineered Scrap[kg] Effective Production Volume[part]
Engineered Scrap per year[kg] =
Number of Cavities[part]
(5.5)
Scrap per Setup[kg] = 5 Shot Weight[kg] (5.6) Material Data
Part Data Process Data Machine Data Mould Data
36 Shot Weight[kg] = Number of Cavities[part] (Part Weight[kg/part]
Engineered Scrap[kg/part])
(5.7)
0, hot runners
Sprue Weight kg +Runner Weight kg , cold runners Engineered Scrap kg/part =
(5.8)This last variable is a function of the feeding system since, with hot runners per example, the material that is injected only forms the parts, with the additional material remaining in the runners to form the next part. Still, in this model, it is established that the number of cavities is equal to the number of injection points, for hot runners, hence the material will be directly injected into the part. For cold runners, the number of injection points is one, forming the sprue and runner system, directly influenced by the number of cavities.
Effective Recycle per year[kg] = minimum (Maximum Recycle per year[kg],
Setup Scrap per year[kg]+Engineered Scrap per year[kg]) (5.9) Maximum Recycle per year [kg] = (Engineered Scrap per year[kg]+
Setup Scrap per year[kg]+Effective Production Volume[parts] Reject Rate[%]
Part Weight[kg/part]) Recycle Rate[%]
(5.10)
The following flowchart, Figure 5.4, is constructed according to the previous equations and the explanation of the PBCM phases, with the intent of demonstrating the influence of the parameters in the material cost. In order to simplify the diagram it was decided to exclude the maximum recycle per year, although it cannot be discarded in the actual model.
Material Cost Reject Rate
Annual Production
Volume
Effective Production
Volume
Number of Cavities
Feeding System
Batch
Material Density
Part Volume
Part Weight
Engineered Scrap
Shot Weight Scrap/Setup
Engineered Scrap/year
Setup Scrap/year
Effective Recycle/
year
Material Unit Cost Material Consumption
Figure 5.4 - Material Cost
Process Model Operations Model Financial Model
37 5.2.1.2 Energy Cost
Another variable cost is energy, which is found according to Equation (5.11) [24].
Energy Cost[€] = Effective Production Volume part ×Energy[kWh/part]
Unit Energy Cost[€/kWh]
(5.11)
According to literature, there are several ways to calculate the Energy parcel of the previous equation.
However, Ribeiro et al. state that the energy consumption can be estimated through an energy balance, which considers the energy to melt the material and fill the mould cavities, Ethermo [J] and the energy related to the machine, Emachine [J]. This energy balance is presented in Equation (5.12) [57].
(-3) machin
thermo e
Energy[kWh/part]=
(E [ J]+E [J ]) 10 × )
3600
(5.12)The first term, Emachine [J] is based on thermodynamic fundamentals and is sensitive to part design, which includes material and geometry, process conditions, including pressure and temperatures, and process efficiency but completely independent from the machine used and disregarding the effect of cycle time. Equation (5.13) [57] clearly demonstrates that it includes the energy to melt the material and to fill the cavities, but includes an efficiency related to the injection machine. Although this is not the energy used to power the machine, included in Equation (5.12), as previously explained, the machine melts the material, through temperature and pressure, and injects the material in the mould cavities. As such, this thermodynamic energy must include a coefficient related to the machine’s efficiency, which, according to literature, is an average value of 80%.
melt fill
thermo
,
E [J]+E [J]
E [J]=
melt fill
(5.13)Now analysing the equation separately, the energy to melt the material is, in theory, dependent on the crystallization degree of the polymer, according to the fundamentals of thermodynamic. Equation (5.14) [57], presents the equation to calculate the melt energy for crystalline and non-crystalline materials, where m is the part weight, Cp is the polymer’s specific heat, Tmelt is the melting temperature, Tamb is the ambient temperature, defined as 20°C, λ is the degree of crystallization and Hf
is the heat of fusion for a 100% crystalline polymer. Although most materials are semi-crystalline, this model accounts for that possibility including the degree of crystallization. All values for these variables, excluding the shot weight, were found in [58, 59].
p
p
melt amb
melt
melt amb F
m[kg]C [J/kg°C](T °C]-T °C]), non-crystalline polymers E [J]=
m[kg]C [J/kg°C](T °C]-T °C])+λ[%]m[kg]H J/kg], crystalline polymers
[ [
[ [ [
(5.14)The energy to fill the mould can be determined as presented in Equation (5.15) [57], by multiplying the injection pressure by the volume of injected material, meaning the part volume.
38
3
fill inj inj
E [J] = P [Pa] V [m× ] (5.15)
Since the feeding system directly influences the necessary volume to be injected, to fill the mould cavity, this term must be divided according to the type of injection, as presented in Equation (5.16). To determine the runner and sprue volume it was considered a cylindrical shape, with the runner diameter to be determined by the designer and the sprue diameter directly influenced by the part weight, as depicted in [60].
3
3 3
part inj
part
V [m Number of cavities[part]+(Runner +Sprue)Volume[m ,cold runners V = V [m /part] Number of cavities[part],hot runners
/part] ]
(5.16)The second term of Equation (5.12), Emachine [J], relating to the energy consumed by the machine, excluding the melting and filling energy, is influenced by the machine type, electric or hydraulic, and the level of installed power, which determines how effective the use of the consumed energy is.
Another contributing factor is the part dimensions, which influence the energy consumption, namely the part thickness, which greatly influences the cooling time for, as explained earlier, a thicker part will need more time to completely cool. Equation (5.18) [57] is chosen to calculate the energy consumed by the machine, including the influence of all these factors through the use of coefficients, where the cycle time can be found in Equation (5.21).
cmachine inst
E = CfM Cf t [s]
[J] P P× [W]
CfT (5.17)
CfM is the machine type coefficient related, as the name indicates, to the type of machine. This coefficient is considered to be 0.5 for electric machines and 1 for hydraulic machines, since studies show that electric machines consume 50% less energy than the hydraulic ones [57].
CfT is the energy thickness coefficient relating the maximum part thickness to the energy consumption and can be calculated according to Equation (5.18) [57].
CfT = 0.0884 s + 0.7629 (5.18)
CfP is the machine power coefficient which shows the fitness of the machine’s nominal power to the part design through the ratio 𝑃𝑡ℎ𝑒𝑟𝑚𝑜
𝑃𝑖𝑛𝑠𝑡 . This ratio relates the thermodynamic power to the machines installed power, translating the suitability of the machine for a specific part, which means that a smaller ratio will indicate an excessive machine dimension, or power, compared to the required to inject the part. CfP can be calculated according to Equation (5.19) [57].
thermo inst
CfP = 1.5079P +0.084
P (5.19)
The nominator of this ratio, thermodynamic power, can be found with Equation (5.20) [57], as the necessary energy to melt and fill the cavities divided by the time for each cycle.
39
thermo thermo
c
[J]
P =E
[W] t
[s] (5.20)
To find the cycle time one must have in mind the previous explanation of the injection moulding process. This indicates that the moulding cycle is composed of three major intervals: fill time, cooling time and open and close mould time (reset time), where this final time is determined as the time the machine needs to open the mould, to eject the part, and the time to close mould again for the next shot. As such the cycle time, Equation (5.21), is the sum of Equation (5.22) [61] and (5.23) [24], and the assumption that, for any setting, the reset time is four seconds, as this is an average value usual for this process.
c
fill time[s]+cooling time[s]+reset time[s]
t =
Number of Cavities [p
[s] art] (5.21)
3 cavity
3 max
2 V [m ] fill time[s] =
Q [m /s]
(5.22)
The Vcavity is the volume of injected material, which is influenced by the type of feeding system as presented in Equation (5.16) [32], and the Qmax is the maximum flow rate of polymer from the nozzle, defined between 100 and 500 [cm3/s] [62].
2 2
2
2 2
D[mm]
ln 0.692Y , cold runners 23.14α[mm
cooling time[s] =
s[mm]
ln kY , hot runners π α[mm
/s]
/s]
(5.23)
Equation (5.23) is divided in cold and hot runner, where, for the former, D is the runner diameter, and for the latter s is the maximum part thickness, k is the part thickness coefficient, determined according to Equation (5.24) [24].
2
4, s 3mm k = π
8 ,s > 3mm π
(5.24)
For both equations that compose the cooling time, α is the thermal diffusivity of the part material, found in [58], which measures the ability of the material to conduct thermal energy relative to its ability to store thermal energy, and Y is determined by Equation (5.25) [24], where Tinj, Tmould and Text are the injection, mould and part ejection temperatures respectively.
inj mould ext mould
Y = T -T
T -T (5.25)
40 Due to the complexity of the energy cost, each parcel of the energy balance, Equation (5.12), is divided into individual flowcharts, where the corresponding sub-models of the PBCM will also be included, to demonstrate the interconnections of all variables composing this parameter. Both flowcharts for thermodynamic and “machine” energy are presented in Annex C.
Although these diagrams appear separately, it is possible to conclude that several parameters influence different variables and, as demonstrated in Equation (5.20), the thermodynamic power depends on the thermodynamic energy. Hence, these parcels of the energy balance, thermodynamic and “machine” energy, are not completely independent.
The final flowchart, Figure 5.5, is constructed considering Equation (5.11), presenting only the two final stages of the PBCM, and implying all relations presented in Figure C.1 and Figure C.2.
Energy Cost Effective
Production Volume
Energy Unit Cost Reject Rate
Annual Production
Volume
Thermodynamic Energy
Machine Energy
Energy
Figure 5.5 - Energy Cost
5.2.1.3 Labour Cost
A final cost considered variable, in this model, is labour, calculated according to Equation (5.26), relating annual paid time, unit cost, and number of direct workers to labour cost [24].
Labour Cost[€] = % direct workers[%]×Annual Paid Time[h]×Cost MH[€/h] (5.26) Although this equation doesn’t demonstrate a direct relation to the annual production volume, a more detailed analysis of its variables validates why this cost is considered variable. As the workers are not dedicated to one product, or machine, the labour cost is considered variable, calculated though the number of workers and hours dedicated to that product.
To calculate the percentage of direct workers, Equation (5.27), is necessary the percentage of line required, Equation (5.28), and the number of workers per machine, defined as 0.25. The percentage of line required is considered as the ratio between the annual required time to produce the defined
Operations Model Financial Model
41 production volume, Equation (5.29), and the line uptime, Equation (5.30), defined as the effective production time in a year [24].
% direct workers[%] = % line required[%]×number of workers per machine (5.27) Annual Required Time[h]
% line required =
Line Uptime[h] (5.28)
Annual Required Time[h] = Effective Production Volume[part]×tc + Effective Production Volume[part]
×Setup Time[h]
Batch×Number of Cavities[par
[h/ r
t]
pa t]
(5.29)
Line Uptime [h] = Working days[days]×(24 - total downtime[h/day]) (5.30) To calculate the line uptime it is necessary to understand the several components of the Total Downtime, as shown in Equation (5.31), where the Paid Breaks are the lunch breaks and other intervals scheduled during a working day, and Idle is the time where the worker is waiting for the machine. Maintenance Time is considered only preventive, and the values are found according to Figure B.1, in Annex B, where a curve is chosen following the indication of complex material, abrasive material and thin features, in accordance with the part design, and the maintenance time determined intersecting the curve with the maintenance level value [32].
Total Downtime[h/day] = Line Shutdown[h/day]+Unpaid Breaks[h/day]
+Paid Breaks[h/day]+Idle[h/day]+Maintenance[h/day] (5.31) As for the effective time to be paid, Equation (5.32), one must consider the number of effective working days per year, including maintenance time and scheduled breaks, for these are still considered a part of the production process.
Annual Paid Time[h] = Working days per year[days] (24 - Line Shutdown[h/day] - Unpaid Breaks[h/day]
(5.32) To find the cost of labour per hour, Equation (5.33), the monthly wages and line uptime must be defined, where 14 is the number of salaries per year and 1.23 is the capital social cost [33].
Wage[€] 14 1.23 Cost MH[€/h] =
Line Uptime[h]
(5.33)
Figure C.3, in Annex C, illustrates Equations (5.26) through (5.33), where the Setup Time and the Reset Time, time to open and close the mould, are assumed to be 30 minutes and 4 seconds respectively.
42 5.2.1.4 Machine Cost
Now focusing on the fixed costs, computed by allocating annual cost, following the financial model for annuity, which is a series of fixed payments, with fixed interest rate, paid at regular intervals, it is possible to find the remaining costs that compose the model.
Equation (5.34) doesn’t present as an annuity but it uses the cost of a machine per hour, which is computed as an annuity allocated throughout the productive time, Equation (5.35), where I is the Investment, n is the machine life and r is the fixed interest rate, defined as 15%. With this result and the annual required time to achieve the necessary volume of production, it is possible to find the annual cost of the machine [24].
Machine Cost[€] = Machine cost per hour[€/h] Annual Required Time[h] (5.34) I×(1-(1+r)-n)/r [€]
Machine Cost per hour[€/h] =
Line Uptime [h] (5.35)
Although there are several possible machines with which to produce a part, it is necessary to guarantee that the chosen machine is capable to manufacture to the necessary specifications, and that the least amount of energy is wasted. To insure this, the clamping force necessary to produce a certain part must be calculated, Equation (5.36) [63], with the number of cavities in the mould, projected area of the part and the pressure inside the mould as variables. As every machine has a maximum clamping force to ensure safe performance, it is necessary to guarantee that the correct machine is used.
i
2
nside mould proj
3
P [bar]×A [cm part]×Number of Cavities[part]
Clamping Force[ton] =
10
/ (5.36)
These variables are dependent on the designer, meaning they are changed directly, except for the pressure inside the mould, which is dependent on the polymer to be used, and the projected area which is dependent on the part design. This pressure must be between 1/3 and 1/5 of the injection pressure, defined as a material characteristic, and in this model it will be considered 20% as depicted in Equation (5.37) [62].
insidemould injection
P [bar] = 20%P [bar] (5.37)
Figure C.4 illustrates the connection between Equation (5.34) through (5.37) and influence of the part, material and mould variables. As it is possible to conclude, this flowchart is very similar to the one of Labour Cost with the major differences being the financial model, where the annuity formula is applied, and the calculation of the clamping force, necessary to establish the machine to be used.
5.2.1.5 Tool Cost
The tool cost is the mould cost allocated throughout the mould life, where I is the Investment, n is the mould life and r is the interest rate, defined as 15%. As the mould life, Equation (5.39), is low if the
43 production volume is high, it is necessary to allocate the investment accordingly, hence the sum of an additional investment in Equation (5.38).
Tooling Cost [€] = I (1-(1+r) )/r + I -n (5.38) If the production volume is low, the mould will last longer. Consequently, in this model, when the mould life, found by Equation (5.39), is higher than the part life, the former will be consider equal to the latter, since the mould cannot last indefinitely.
Mould Life [shots]
Mould Life [years] =
Effective Production Volume [part]
Number of Cavities [part]
(5.39)
To find the mould investment, Equation (5.40), several costs must be considered, where only the feeding system cost and structure cost are variable. This is in response to fact that this model only accounts for the type of feeding system and the number of cavities, hence influencing the mould dimensions.
Mould Investment [€] = Structure Cost[€]+Acessories Cost[€]+
Feeding System Cost[€]+Manufacturing Cost[€]+Other Costs[€] (5.40) In this model, only the structure, feeding system and manufacturing costs will depend on process variables, meaning the dimensions of the part, the type of feeding system and the time to p roduce the mould, respectively. This manufacturing cost is found by multiplying the number of hour needed to produce the mould, for each manufacturing process, by the cost per hour of each process. The number of hour is an input defined for one cavity and multiplied by the number of cavities defined.
Although the additional costs are important, they are considered constant, where the accessories cost is 812€ and the other costs, meaning the raw material costs, polishing, etc. are 992€. These values were taken from a previous model [32].
To calculate the amount of material to produce the mould, Equation (5.41), it is necessary to first find the volume of the mould, Equation (5.42).
3 3
Mould material consumption[kg] = Mould Volume[cm ]
[kg/cm ] (5.41) Mould Volume [mm = (Part height+30)[mm] 2+(Mould Height-100)[mm]+3(Mould Width-100)[mm]
]
(5.42)
Figure 5.6 illustrates the correlation between Equations (5.38) to (5.40) and the PBCM levels.