Which color of an umbrella is more effective against sunlight? It's usually said that black umbrellas are best against sun, since black absorbs most of the radiation . The common umbrellas in market(atleast in India) are painted black outside and silvery inside. 
However I think the reverse(silvery outside, black inside) should be more effective. The outer color(silvery) will reflect most of radiation, while the black color inside will absorb anything passing through outer layer.
Now my questions are :
   1. Which color scheme is more effective ?
   2. What is the logic behind the common (black-outside, silvery-inside) scheme ?

 A: Even considering the same fabric with different colours it will depend a lot on optical properties of the dye which we cannot tell only by the colour that we can see. I would need to know "how much the silver paint would reflect and in what spectral range" and also "how is the absorption coefficient as a function of the wavelength (from near-UV, Visible to near-Infrared)". I believe that this is the most important point and not the colour that we perceive with our eyes.
Nevertheless, keeping things simple, I think that the reflective coating being outside or inside would make the same amount of radiation that passes through. The only difference is that putting the reflective layer inside you would be increasing the path length of the radiation, thus increasing the absorption, and therefore the temperature
EDIT:

In order to clarify my idea I added this picture. Dashed layer is reflective and the the grey layer is the normal dark layer. I would like to divide this discussion in two: "Radiation Protection" and "Heat Protection"
Regarding the radiation protection, I believe that it depends on how the the dark layer absorbs light in all spectral range of the sun light, i.e. absorption coefficient and thickness of the material+dye. It also depends on how the reflective layer reflects the light, i.e. reflectance. I believe that the radiation protection does not depend on the arrangement, so both schemes protect from radiation as good. (I am not considering far infrared radiation)
Regarding Heat Protection, the thought is the same, except we should consider that in the first case the length of the radiation in the dark layer is longer, thus absorbing more radiation thus heating more. Also, the protection from the heat will depend on the heat conductivity of the layers and surface morphology which will affect how well the layer will cool down.
I understand that having a reflective coating outside an umbrella would be unpleasant for other people. But it would be the best choice for heat protection. However, regarding the emergency heat blankets, I do not know why they also have a reflective layer on the inside.
A: This has to do with emmitance not reflectance.  The reflective surface on the inside prevents heat from being emmitted toward the user from the hot umbrella.  The black surface..while having a low reflectance...has a high emmittance.  I think they must have found that the black surface (with the high absorbance) prevent radiation from just passing through the umbrella then lets convection keep the umbrella cool.
edit for clarity
The inverse of reflectivity is emmittance under steady state.  If a surface is reflective, it will not emit much radiative heat.  If a surface is black (or any dark color), it will absorb heat and then emit it.  When the light hits a material that is thin and reflective on both sides, light will reflect and some will pass through (making it thick will reduce the amount that passes through). When the light hits a material that is thin and dark on both sides,  the material will absorb the heat and then emmit it (roughly) evenly through both sides.  If, however, the material is dark on the outside and reflective on the inside.  The material will absorb through the dark side but then only emit through the dark side.  This will result in the individual within the umbrella not feeling the heat from the radiation.
This is basic physics and is used for roofs, thermoses, spacesuits, and, apparently umbrellas but i've never seen them.
A: I would say that the most reflective surface has to go outside; then the one that absorb the most, which is the one that will get more heat, as "cinico" said.
So, first the most of the radiation bounces off, then the rest, which should be small, is absorbed by the black fabric.
Take into account anyway that we are talking about the effect of reflectivity only. It could happen that the fabrics have different specific heat, so one would convert radiation into temperature faster than the other. 
