What are the factors affecting Bouyancy and Archimedes Principle?

Archemedes principle is the principle from which the concept of buoyancy was generated.
Archemedes principle is simply a statement and there cannot be any "factors" affecting it. It states that an object immersed in a liquid experiences a buoyant force which is equal to the weight of the liquid displaced by it. eg. If you enter a tub full of water, some water will come out. Now, if you calculate the weight of the water that comes out, you will get the value of the buoyant force.

The phenomenon of you experiencing that buoyant force is called buoyancy. Please refer to the given link for factors affecting buoyancy:
https://www.meritnation.com/ask-answer/question/defination-for-buoyancy-and-factors-affecting-magnitud/gravitation/1523691

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There are multiple factors that affect how much will the buoyant force act on an object. In this article we shall discuss those factors and have an idea about this buoyant forcei) For example, when a cork is pushed down into water kept in a glass, the cork rises on its own and comes back to the surface. The reason is that the pressure pushes cork upwards to keep it beneath the surface of water, downward force need to be exerted. This shows that, when the cork is under water, despite its weight, there is some force which pushes it upwards. When it is under water, the upward push is a kind of force exerted by water. The downward force exerted counter acts this upward push of the water on the cork.The ability of a liquid to exert an upward force on an object placed in it there by making it float or rise is called buoyancy.

ii) The buoyancy can also be seen when we are swimming, we float on water. This is also due to the upward force exerted by water on our body.

iii) A man can lift a heavier stone more comfortably under water than he could in air. This is because of the upward force of the water reduces his effort.

In all the three cases, when the objects are in water they experience a lifting force within the water and this force is called up thrust.

The force of the up thrust is equal to the weight of liquid displaced by the body. This is also called as Archimedes principle.

Major Factors that Affect Buouancy Force

Buoyancy of a body depends on the following factors:

Volume of the body submerged in the liquid or volume of the liquid displacedDensity of the liquidAcceleration due to gravity

Volume

When we push a small piece of cork into water, we experience some up thrust on our fingers. Suppose if we push a bigger piece of cork into water, we experience a greater up thrust on our fingers. Thus, it can be said that the buoyancy force increases with increase in the volume of the object.

Density

When we push a small piece of cork into water, we experience some up thrust on our fingers. Let us add some salt to the water and repeat the same experiment. It can be observed that we experience more up thrust on our fingers. With this it can be concluded that buoyancy increases with increase in density.

Acceleration due to gravity

The force of buoyancy also increases with increase in acceleration due to gravity.Effects of Factors Affecting the Buoyancy Force

Let us now consider the overall object density. If it is less than that of the fluid, the object will float and achieve the above condition once it comes to rest. On the other hand, if the overall object is greater than that of the fluid, it will sink. Even when the body sinks, buoyant force acts on it. Just that it is dominated by the force of gravity acting on the body. When in the process of sinking or already sunk, the buoyancy force depends on the volume of the object, the density of the fluid and more importantly, the local gravitational field