Description of the cost factor development value

In the model, we make use of a cost development factor, cf, which is used to describe future costs of AM. Since Thales, will outsource the AM processes, the costs of AM can be translated into the variable production costs per piece. Based on current trends and expert opinions, we can conclude that the costs of AM will decrease in future. However, to what extend and how fast this decrease will be, is highly uncertain.

According various researchers and experts, the costs will decrease as a result of economies of scale. In the future, the market of AM will grow. This will result in lower prices of materials used for different AM processes and the production costs of machines will decrease. Additionally, the build rate of AM

processes will increase in the future, resulting in more efficient production processes. Experts expect that the costs of AM will drop by 50% within the next five to ten years. Dr. Bernhard Langefeld from Roland Berger Strategy Consultants, carried out a research in 2014 commissioned by VDMA about the expected future cost development of AM. In his research Langefeld (2014), states that during the period from 2013 to 2018 costs can be cut with 49%. Additionally, he states that during the period from 2018 to 2023 costs can be cut with another 32% (in total a decrease of more than 65%). Although we have little information, based on these statements we can conclude that the costs of AM in the near future (five to ten year) will decrease faster than the costs of AM in the far future (more than ten year). This is in line the expected cost development according AM experts. Such a decrease can be displayed by an

exponential function, such as:

𝑐𝐴𝑀(𝑡) = 𝑐𝐴𝑀(0) ∗ 𝑐𝑓𝑡

Where,

𝑐_𝐴𝑀(𝑡) = variable production costs of the AM method per piece in period t.

𝑐_𝐴𝑀(0) = variable production costs of the AM method per piece in period 0.

𝑐𝑓 = cost factor to describe future cost developments for the variable production costs per piece of the AM method.

𝑡 = period of the life cycle where are taken into account.

Such a formulation however, ensures an extreme decrease of costs in the far future. If we take into account a realistic decrease in the near future as opposed by Langefeld of parts demanded by Thales which have to meet high requirements, the values will become unrealistically low at a certain point, see Figure 18. People within Thales assume that the minimum costs after 50 years will be somewhere between 15%-25% of the costs at this moment in time. Therefore, we have sought a more appropriate function to model the cost development of AM.

We will use the ‘learning curve’ to model the cost development in the future. A learning curve is a

mathematical model to express the phenomenon of increased efficiency and improved organizational performance with repetitive production of a good or service (Sullivan, Wicks, Koelling (2015). The concept is that some input parameter decreases, on a per-output-unit basis, as the number of units produced increase. Most learning curves are based on the assumption that a constant percentage reduction occurs as the number of output doubles. In our case, however, we won’t model on a per- output-unit basis, but on a per-period basis and thus we assume a constant reduction of cf % as the number of periods doubles. The function used is as follows:

Where 𝑛 =log 𝑐𝑓_{𝑙𝑜𝑔2} = the learning curve exponent

𝑐𝑓 = % reduction of the variable production costs per piece of the AM method every time the analysed period is doubled

𝑡 = is the period of the life cycle analysis

The reason, we take a per-period basis is that we have to make decisions at the beginning of a period for the rest of that period. Cost adjustments during a period will be translated into an average number resulting in another type of function rather than a complement to the cost function. The value for cf can be determined by solving the following equations, where x is de % cost decrease of AM over t periods:

𝑐_𝐴𝑀(𝑡) = 𝑐𝐴𝑀(0) ∗ 𝑡𝑛 (1 − 𝑥) = 𝑡𝑛 (1 − 𝑥) = 𝑡𝑛 log𝑡(1 − 𝑥) = log𝑡(𝑡𝑛) 𝑛 = log_𝑡(1 − 𝑥) 𝑛 =ln (1 − 𝑥) ln (𝑡) In order to calculate cf: 𝑛 =log(1 − 𝑐𝑓) log 2 log(1 − 𝑐𝑓) = 𝑛 ∗ log 2 𝑐𝑓 = 1 − 2𝑛

By filling in n, in the previous expression we get:

𝑐𝑓 = 1 − 2ln (1−𝑥)ln 𝑡

Now, we can calculate cf based on an expected decrease of the costs of AM during a certain period. Value of the cost development factor

Based on the numbers provided by Langefeld (2014), we are able to determine a value which could be used as cf. This number is based on the period from 2018 until 2023 in which the costs will decrease with 32%. So, x will be 32% and t = 5 periods:

𝑐𝑓 = 1 − 2ln (1−0.32)ln 5 _{= 15,3%}

If we fill in this value in the function, we get the following line as presented in Figure 18. In consultation with several people within Thales, we conclude that this development curve is quite a conservative one. The experts within Thales who are dealing with AM in some way, expected that the production costs of AM will decrease by 50% within the next 5 to 10 years. Therefore, we also plotted the cost curve presenting a reduction of 50% in 5 years and the cost curve presenting a reduction of 50% in 10 years in Figure 18 in order to compare them easily.

Decrease of 50% in 5 years

𝑐𝑓 = 1 − 2ln (1−0.5)ln 5 _{= 25,8%}

Decrease of 50% in 10 years

𝑐𝑓 = 1 − 2ln (1−0.5)ln 10 _{= 18,8%}

Figure 18: cost development curves

Based on these curves, and the expectation that costs after 50 years will be somewhere between 15%- 25% of the costs at this moment in time. We conclude that the cost development will be described in the best way if we take cf 25,8%. This corresponds with a decrease of 50% over 5 years from now.