## 163 Cost modeling

The manufacture of a foam (or of a component made from one) consumes resources (Table 16.1). The final cost is the sum of these resources. This resource-based approach to cost analysis is helpful in selecting materials and processes even when they differ greatly, since all, no matter how different, consume the resources listed in the table. Thus the cost of producing one kg of foam entails the cost Cm (\$/kg) and mass, m, of the materials and feedstocks from which it is made, and it involves the cost of dedicated tooling, Ct (\$), which must be amortized by the total production volume, n (kg). In addition, it requires time, chargeable at an overhead rate CL (with units of \$/h or equivalent), power P (kW) at an energy cost Ce (\$/kW.h), space of area

 Resource Symbol Unit Materials: inc. consumables Cm \$/kg Capital: cost of equipment Cc \$/unit cost of tooling Ct \$/unit Time: overhead rate Cl \$/hr Energy: power P kW cost of energy Ce \$/kW.h Space: area A m2 cost of space Cs \$/m2.h Information: R&D Ci royalty payments

A, incurring a rental cost of CS (\$/m2.h), and information, as research and development costs, or as royalty or licence payments C; (expressed as \$/h). The cost equation, containing terms for each of these, takes the form

[Material] [Tooling] [Time] [Energy] [Space] [Information]

rcfi

Cl

PCe

aCs

Ci

C

C

C

C

n

n

n

n

where ni is the production rate (kg/h) and f is the scrap rate (the material wastage in the process).

A given piece of equipment - a powder press, for example - is commonly used to make more than one product, that is, it is not dedicated to one product alone. It is usual to convert the capital cost, Cc, of non-dedicated equipment, and the cost of borrowing the capital itself, into an overhead by dividing it by a capital write-off time, tc (5 years, say) over which it is to be recovered. Thus the capital-inclusive overhead rate becomes

[Basic OH rate] [Capital write-off]

where CL0 is the basic overhead rate (labor, etc.) and L is the load factor, meaning the fraction of time over which the equipment is productively used.

The general form of the equation is revealed by assembling the terms into three groups:

[Material] [Tooling] [Time Capital Energy Space Information]

The terms in the final bracket form a single 'gross overhead', CL,gross, allowing the equation to be written

The equation indicates that the cost has three essential contributions: (1) a material cost/unit of production which is independent of production volume and rate, (2) a dedicated cost/unit of production which varies as the reciprocal of the production volume (1/w), and (3) a gross overhead/unit of production which varies as the reciprocal of the production rate (1/n). Plotted against the production volume, n, the cost, C, has the form shown in Figure 16.2. When the production volume, n, is small, the cost per kg of foam is totally dominated by the dedicated tooling costs Ct. As the production volume grows, the contribution of the second term in the cost equation diminishes. If the process is fast, the cost falls until, often, it flattens out at about twice that of the constituent materials.

Production volume

Figure 16.2 The variation of material cost with production volume Technical cost modeling

Production volume

Figure 16.2 The variation of material cost with production volume Technical cost modeling

Equation (16.3) is the first step in modeling cost. Greater predictive power is possible by introducing elements of technical cost modeling or TCM (Field and de Neufville, 1988; Clark et al., 1997), which exploits the understanding of the way in which the control variables of the process influence production rate and product properties. It also uses information on the way the capital cost of equipment and tooling scale with output volume. These and other dependencies can be captured in theoretical and empirical formulae or look-up tables which are built into the cost model, giving greater resolution.

The elements of a TCM for liquid-state foaming of aluminum

Consider a cost model for the production of panels of a SiC-stabilized aluminum-based metallic foam by the process illustrated in Figure 16.3 and described in Chapter 2. There are four steps:

Melting Holding Foaming Delivery

Schematic of the liquid-state foaming process.

Melting Holding Foaming Delivery

Schematic of the liquid-state foaming process.

1 100 200 300 400 500

Production volume (Thousands of units/Year)

Figure 16.3 Schematic of the liquid-state foaming process

1 100 200 300 400 500

Production volume (Thousands of units/Year)

Figure 16.3 Schematic of the liquid-state foaming process

• Melting of the pre-mixed alloy

• Holding, providing a reservoir

• Foaming, using compressed gas and a bank of rotating blades

• Delivery, via a moving belt

The output of one step forms the input to the next, so the steps must match, dictating the size of the equipment or the number of parallel lines in each step. The capital and tooling costs, power, space and labor requirements for each step of the process are cataloged. Data are available characterizing the dependence of material density and of production rate on the gas flow rate and the stirring rate in the foaming step. These empirical relationships and relationships between equipment and production rate allow the influence of scale-up to be built into the model.

The outputs of the model (Figure 16.4) show the way in which the cost of the material depends of production volume and identifies cost drivers. Significantly, the model indicates the production volume which would be necessary to reach the plateau level of cost in which, in the best case shown here, the cost of the foam falls to roughly 2.5 times that of the constituent materials. Models of this sort (Maine and Ashby, 1999), applied to metal foam production, suggest that, with large-volume production, the cost of aluminum foams made by the melt foaming method (Section 2.2) could cost as little \$3/lb; those made by the powder route, about \$6/lb.

206 Metal Foams: A Design Guide PV = 20,000 units/yr

- Materials cost

- Direct labor Energy cost

■ Equipment cost -Auxillary Tooling cost Maintenance

Fixed overhead Building cost Working capital

- Materials cost

- Direct labor Energy cost

■ Equipment cost -Auxillary Tooling cost Maintenance

Fixed overhead Building cost Working capital

 PV = 300,000 units/yr 11 —'—' - - 3 !=, - - - !? _

"Liquid n PM, batch n PM, continuous

Figure 16.4 Output of the TCM for the liquid-state foaming process of aluminum

We now consider the final step in determining viability: that of value modeling.

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