Most footpaths were used for transporting goods, and therefore it is appropriate to combine the slope cost component with a component depending on the load. It is general knowledge that energy expenditure becomes greater with increasing loads. However, variation in energy expenditure depending on slope and load is high. Minetti *et al*. (2006) refer to studies of loaded walking which showed that on level terrain, both Nepalese porters and East African women use remarkably less metabolic energy than Caucasian control subjects. In their figure 1 they present the walking speeds of Nepalese porters loaded with 45kg along paths as a function of gradient. The variation of the speed is quite large; on level ground most speed measurements are in the range of 0.9 to 1.3 m/s.

According to Knapik (2001), until about the 18th century the load carried by soldiers (including Greek hoplites and Roman troops) seldom exceeded 15kg while they marched. According to Roth (1998, 9) male adults in antiquity were certainly shorter than today, and based on skeleton data from northern Europe, Steckel (2004) concludes that the average height of men decreased from 173.4 cm to 167 cm during the seventeenth and eighteenth century. It may be due to the taller stature that modern soldiers are able to carry heavier loads. Knapik argues that overloading troops can lead to excessive fatigue and impair the ability to fight. A similar argument may hold for prehistoric transport. Carrying very heavy goods for a long period of time impairs the health and in the long run it is more effective to carry smaller loads. Langmuir (2004, 247) states that the loads of a mountaineer should never exceed one-third of the body weight of the individual. But the example of the Himalayan porters, who are smaller than European mountaineers, tells another story. As mentioned above, Nepalese porters start carrying about 35 kg at the age of ca.12 years (Minetti et al. 2006). Minetti and his co-authors explain the higher metabolic power of porters by training, anatomy and adaptation of this ethnic group to the challenging environment. Knapik (2001) notes that walking with backpack loads over a period of weeks results in a decrease in the energy cost of carrying the load. Maybe Nepalese porters are an extreme example of decreased energy cost due to training. However, accounts for Mesoamerica also refer to bearers carrying loads ranging from 20 to 50kg, and Drennan (1984) uses a standard load of 30kg for Aztec bearers in his calculations. As the ethnographic and the historic data concerning the carrying capacity of a walker vary, it seems a good idea to compare the results of several LCP calculations based on different load values.

Moreover, Knapik (2001) describes twelve ways to carry loads, including backpacks, head basket, trunk vests, transverse and front shoulder yokes. None of the load transport methods presented involves high technological skills, so all of them might have been used by prehistoric people. Physiological studies show that loads are carried most effectively on the head, which requires long training, is useful only in unobstructed horizontal terrain, and produces a high profile, i.e. it is difficult to hide with loads on the head. The impact of gender is significant, even when differences in body size and body composition are taken into account. Minor injuries such as foot blisters can adversely affect an individual's mobility and walking effectiveness.

The models presented in this section were developed by physiologists who are experts in measuring energy expenditure in humans and animals. Givoni and Goldman (1971) present the following formula for predicting energy costs (kcal/h):

`Cost(N,W,L,V,s) = N*(W+L)*(2.3 + 0.32*(V-2.5)`

^{1.65} + s*(0.2+0.07*(V-2.5)))

with N the terrain factor, W weight in kg, L load in kg, V velocity in km/h, and s per cent slope. The terrain factor is 1 for treadmill walking. For all given velocities, terrain factors and gradients, the energy expenditure estimate given by the above formula is proportional to the total weight moved. This agrees with the result of the study by Yousef *et al*. (1972), which showed that the net energy expenditure per kilogram is the same for load or no load. Multiplying the cost function by a constant value does not change the LCP results. Therefore the following simplification is appropriate for LCP purposes:

```
Cost(N,V,s) = N*(2.3 + 0.32*(V-2.5)
```

^{1.65} + s*(0.2+0.07*(V-2.5)))

According to Givoni and Goldman (1971), the formula is valid for walking speeds from 2.5 to 9 km/h, for grades up to 25%, and loads up to 30kg carried near the centre of gravity of the body.

In most physiological studies, the number of people tested is fairly small and mostly confined to men. For example, the study by Minetti *et al*. (2002) measured the energy expenditure of ten men with an average age of 32.6. The study by Givoni and Goldman pools the results of several series of experiments also involving women. So the formula is probably more reliable than that derived from the Minetti *et al*. data – in the uphill slope range of 0 to 25%. The main drawback of this formula is the limited slope range and the dependence on the velocity variable. Most people adjust their velocity depending on slope; unfortunately no formula for preferred or optimised walking speeds is included in the study. Assuming a constant velocity of the walker is not realistic and results in a linear slope-dependent cost function, the drawbacks of such a function were presented above. It seems that no archaeological LCP study applied this formula.

The initial formula by Givoni and Goldman (1971) was adjusted by Pandolf *et al*. (1977) to account for slow speeds. The authors suggest the following formula for energy expenditure in Watt:

`Cost(N,W,L,V,s) = 1.5*W + 2.0*(W+L) * (L/W)`

^{2} + N*(W+L)*(1.5*V^{2} + 0.35*V*s)

with N, W, L, and s as above but the velocity of walking (V) is given in m/s. This formula assigns a higher cost to walking on steep gradients at speeds in the range of 1.0 to 2.0 m/s (2.8–5.6 km/h) than the formula published previously. In the absence of an external load, the energy expenditure estimate given by this formula is proportional to the weight of the subject. The energy expenditure in Watt per kg for a walker carrying no load is:

`Cost(V,s) = 1.5 + N*(1.5*V`

^{2} + 0.35*V*s)

Contrary to the Givoni and Goldman formula, the result of the Cost(W,L,V,s) formula is not proportional to the load and weight total; instead the relation of the load to the body weight plays an important role. The study by Duggan and Haisman (1992) found when comparing various models for uphill and level walking with new experimental data that the formula by Pandolf *et al*. (1977) fitted well. However, only one walking velocity (1.67 m/s) was tested and the grades were in the range of 0 to 6%. According to McArdle *et al*. (2006, 213), the relationship between walking speed and energy expenditure remains approximately linear between speeds of 3 and 5 km/h (i.e. 0.83 to 1.39 m/s); at faster speeds, energy expenditure increases disproportionally. It can be shown that the estimates provided by the Pandolf *et al*. formula show the same trend.

Both Branting (2004, 74) and the GIS book by Wheatley and Gillings (2002, 155) refer to this formula. Murrieta Flores *et al*. (2009) apply the formula converted to kcal/hour. They assume a constant speed of 5 km/h but do not give any details about the choice of the other parameters in the Pandolf formula. Van Leusen (2002, chapter 16, 12–15) uses this formula for his Wroxeter hinterland study but confuses the units of velocity, i.e. V is measured in km/h instead of m/s. Van Leusen applies the formula for given fixed values of W, L, N, and V, and in this case, the formula is linearly dependent on per cent slope. The disadvantages of such a linear cost function were discussed in Section 5.1.4.1.

In another chapter (chapter 7, 6) van Leusen modifies the Pandolf formula with the aim of creating a function with a minimum at a slope of about 6° (i.e. 10%) downhill, but due to confusion of units yet again the minimum is at 6% on a downhill gradient:

`Cost(N,W,L,V,s) = 1.5*W + 2.0*(W+L) * (L/W)`

^{2} + N*(W+L)*(1.5*V^{2} + 0.35*V*|s+6|)

with N, W, L, V and s as in the formula by Givoni and Goldman. Ejstrud (2005) as well as Conolly and Lake (2006, 220) copy the above formula including all mistakes from van Leusen. Even with correct measurement units, this adjusted formula is no longer in agreement with the data found in Pandolf *et al*.'s experiments.

With the parameters chosen by van Leusen, the Pandolf formula estimates that climbing a slope of 10% requires about twice as much energy expenditure as on level ground. However, the incorrect formula implies that energy expenditure doubles at a slope of more than 20% compared to flat terrain – and this is quite a difference!

Even if the Pandolf formula is applied correctly, the disadvantage remains that it does not model how walkers adjust their velocity to slope and to the load they are carrying (cf. Todd and Scott 2002). It is possible to combine the Tobler hiking function with the Pandolf cost formula. However, based on the formula modified by van Leusen, the energy expenditure decreases with increase of slope! Obviously this strange result is due to the confusion of the velocity units by van Leusen.

The formula of Pandolf *et al*. can be used to determine the cost of walking 1m at a given speed on a given gradient s:

`Cost(W,L,V,s) = (1.5*W + 2.0*(W+L) * (L/W)`

^{2} + N*(W+L)*(1.5*V^{2} + 0.35*V*s))/V

Calculating the minimum of this cost curve with respect to variable V with given values of W, L and s determines the optimal speed of walking. After some algebraic transformations the optimal speed is given by the following formula:

Note that this formula is independent of slope, i.e. the optimal walking speed is only dependent on the weight and the load. This does not agree with everyday experience. Moreover, the optimal speed for an unloaded subject is 1/√N m/s, regardless of the subject's weight.

As a conclusion from the findings above, a subject without external load walking at optimal speed (1 m/s) on a blacktop road expends the following amount of energy (kilo-joule expended for each kg of the walker covering 1m):

`Cost(s) = 3 + 0.35*s`

The disadvantages of linear cost functions were discussed above, and in general, the Pandolf *et al*. formula does not seem optimal from a theoretical point of view. Branting (2004, 76) also points out several drawbacks of the Pandolf *et al*. model: It is designed to cover positive slopes only; the experiments to test the formula cover the small range of 10% downslope to 25% upslope, whereas the grades in the study areas of most LCP analyses exceed 25%. Moreover, the estimate of energy expenditure becomes negative for steep downslope gradients, i.e. the estimate is negative for a subject weighing 70kg walking down a slope of 8.6% at a speed of 1 m/s.

Moreover, the approach by Pandolf *et al*. (1977) has been criticised for not being able to differentiate between different modes of carrying loads (Laursen *et al*. 2000). Head-supported carrying is considered most cost-effective, and carrying loads asymmetrically requires significantly (about 10%) more energy than symmetric loads. The energy expenditure of walking with symmetric loads (0, 10 and 20kg) was recorded in experiments by Laursen *et al*. (2000) with six healthy men (age: 29–36 yr, weight: 72–89kg). These measurements showed an unexpected result. Walking on level ground the energy expenditure depended only on the total weight (weight + load) as suggested in the Givoni and Goldstein formula and not on the load to weight ratio as in the Pandolf formula.

In conclusion, it does not seem appropriate to apply the Givoni and Goldstein or the Pandolf formula for archaeological LCP calculations mainly owing to the limited slope range, but other drawbacks have also been found. Similarly, experiments with a limited slope range of 0 to 12% form the basis of the formulas proposed by Santee *et al*. (2003), and therefore these too should not be used in archaeological LCP studies.

Branting (2004, 74–7) suggests using another cost model based on physiological studies on several hundred male and female subjects. The formulas forming this cost model cover a wider range of slopes (-40% to 40%), but also include a velocity variable, and estimates for this variable are not included in the publication by Branting. According to one of the formulas presented in this context, energy expenditure depends on the weight of the individual including all weights carried, and not on the load to weight ratio.