I am planning to bring my family members to visit London for a vacation of 20 days. If I am taking full sponsorship including accommodation, food, and travelling, how much amount do I need to show for a person?
With the permission of @GayotFow, as the editor of his currently-in-draft canonical, here is what you should understand about sponsorship. In this excerpt note the first statement: your sponsorship does not mean that the applicants don't have to show that they qualify for a visa, they do.
Refused applicants have often made the assumption that having a sponsor eliminates the need to demonstrate that they qualify. This is not true and, in many cases, needing a sponsor actually increases the onus on the applicant to demonstrate that they qualify.
How can this be? To understand this paradox, return to Paragraphs (a) and (c) again and note that neither of them is contingent on having access to money. Indeed, having access to money is covered in Paragraph (e):
must have sufficient funds to cover all reasonable costs in relation to their visit without working or accessing public funds. This includes the cost of the return or onward journey, any costs relating to dependants, and the cost of planned activities such as private medical treatment.
...which is rarely mentioned at all in most refusals! So it's not about the money and it doesn't matter how wealthy the sponsor is or how much money they are willing to commit; the onus never leaves the applicant to demonstrate that they qualify. See UK visitor visa refused (multiple sponsors) for an example which says:
This refusal cites Paragraphs V 4.2 (a) and (c) of the rules... You can see that they did not challenge your daughter's capacity for sponsorship, they accepted her sponsorship without question. But having a credible sponsor does not alleviate the applicant's burden of meeting the rules.
The ECO is also entitled to be mindful of the sponsor's relationship to the applicant and why the sponsor is willing to undertake a large expenditure of no apparent benefit to them. This is especially true, for example, if the applicant has siblings who have never had sponsorship. When there is no history of a grandparent or parent sponsoring other family members and a single person is going to benefit, the motivation should be carefully explained.
Finally, attestations by the sponsor along the lines of "...my family very much respects UK law and would never overstay..." can damage the sponsor's credibility. Nobody is telepathic and telepathy is the only way such an attestation can be made. See Applied for Family Visit but refused under V4.2(a) and (c):
It doesn't cut any ice to assure them that you will return home after your visit, and assurances like that make for a particularly weak application anyway. Why should you return home?
This is stated more formally in My girlfriend was a refused a visa to visit me in the UK. What now?
A sponsor's responsibility is to demonstrate the capacity to maintain and accommodate the application. Nothing else. Your 'intentions' are irrelevant and attempting to assert them indicates that you do not understand what your role is and, by extension, that you do not understand the rules. This is a fair assumption on their part because, if your friend decided to go underground and disappeared inside the UK, there is nothing you could do to prevent it.