Does positive total entropy generation means that the entropy generation of the system and the entropy generation of the surroundings are positive too When I was working on problems related to entropy generation, there was this problem that asks if the process claimed in the problem is possible or not. The solution manual calculated the total entropy change and it was positive, therefore the total entropy generation is also positive, so it concluded that the process is possible (occurs spontaneously).
But I was thinking, the total entropy generation is the sum of the entropy generation of some systems, then if the total entropy generation is positive this doesn't mean that the entropy generation of each system is positive, and the process that was concluded possible may not be possible. Shouldn't we inspect for each system separately. 


To make the question clearer, this is a derivation of the principle of the increase of entropy from my text book. At the end of the derivation (equation 6.39), you see that total entropy change equals to S(gen A) + S(gen B)+ S(gen C). Now if we inspect the total entropy change and it was positive, does this necessarily mean that S(gen A) is positive and S(gen B) is positive and S(gen C) is positive, shouldn't we inspect for each system separately.  
Can any one help me with this? Thanks in advance.
 A: It is not clear what you mean by "external entropy generation is due to heat transfer through finite temperature difference."  This portion of the entropy generation is included either in the entropy generated in the system or the entropy generated in the surroundings (or both).  However, the surroundings is usually modeled as an ideal reservoir (or reservoirs) in which case the entropy generation in the surroundings is zero.  Therefore, this portion of the entropy generation is typically part of the entropy generated in the system.
With regard to the second part of your question, entropy generation is always positive, no matter where it occurs.  However, entropy transfer as a result of heat flow from one entity to another can be either positive or negative, depending on whether the heat is flowing in or out.  So, there are two ways that the entropy of an entity can change:  entropy transfer from another entity or entropy generation (within the entity under consideration).
ADDENDUM
In their book Transport Phenomena by Bird, Stewart, and Lightfoot, they prove in Chapter 11, Example 11D.1., Equation of change for entropy, that the local rate of entropy generation per unit volume in a system experiencing an irreversible process is the sum of three contributions: one contribution proportional to the squares of local velocity gradients, a second contribution proportional to the square of the local temperature gradient, and a third contribution proportional to the squares of the local species concentration gradients.  Since squares of the quantities are involved, they are all positive definite.
A: Foundations
Let's first define a process as a change in a system from an initial state to a different final state.
The term "possible" is perhaps one point of confusion. Any process can be proposed to be a possible process. Only certain such proposed processes are however spontaneous. They will occur with no external input. Other processes are not spontaneous.
The second important point of clarification distinction between $S_{gen}$ and $\Delta S$ for any process. The $\Delta S$ for any process in any one location (control volume, system, or surroundings) includes two terms. One is the reversible entropy change $\Delta S_{rev}$ of the process. The other is the irreversible entropy change $\Delta S_{irr}$ of the process. In engineering applications, the latter term is also called the entropy generation $S_{gen}$. A further discussion of the meaning of these two terms is found at the answer for this question.
Now to the key point. Irreversible entropy generation is always a positive value. We do not define an irreversible process for a given control volume or system or surroundings to generate a negative irreversible entropy. An insight to this is also that we use $S_{gen}$ rather than $\Delta S_{irr}$ specifically because the latter form with a $\Delta$ might suggest that irreversible processes could generate a positive or negative irreversible change, and this is not true.
The entropy criteria that is used to determine whether a process is spontaneous must use the TOTAL entropy change $\Delta S_T$. This is typically called the entropy change of the universe. It is the sum of the entropy change of the system and the surroundings $\Delta S_T = \Delta S_{sys} + \Delta S_{surr}$. Each term for the system or surroundings $\Delta S$ has both reversible entropy change and irreversible entropy change (generation).
When the total entropy change of any proposed process is positive, the process will be spontaneous as proposed. When the system is left to its own, the preferred direction is for it to go from the initial state to the proposed final state. Spontaneous processes are not associated with the direction of heat flow to/from the system. However, they are associated with the direction of heat flow from hot to cold. Ice melts spontaneously at room temperature because heat flows into it. The entropy of the ice increases as it goes to water. Water freezes at temperatures below the freezing point because heat flows out of it. The entropy of the water decreases as it goes to ice. In both cases, the total entropy change of the universe is positive. In both cases, heat spontaneously flows from hot to cold temperature.
When the total entropy change of a proposed process is zero, the process will be at equilibrium at all times throughout the process. The entropy exchanged between the system and the surroundings are exactly equal and opposite. The proposed change from the process will not occur spontaneously. However the process can be driven to occur, and in this case, the process follows a reversible path.
Finally, when the total entropy change is negative, the process will not be spontaneous as proposed. In this case, the inverse process will be spontaneous.
In summary, during any proposed (and therefore hypothetically "possible") process, each entropy component for the system and surroundings can be positive, negative, or zero. The SUM defines whether the proposed process is spontaneous or not.
Specific to Your Example
The example you give has three locations: A, B, and C. In the picture, location C is common to A and B. We would intuitively define C as the surroundings to A and B. For us to state that a process that occurs in this universe is spontaneous, the total entropy change of this universe of system A + system B + surroundings C must be greater than zero. The entropy changes for the separate processes that each occur in system A, system B, and surroundings C can individually be positive, negative, or zero. When the sum is zero, the three locations are in equilibrium. When the sum is negative, the proposed process is not spontaneous as written.
The specific example you post ends with only a sum of three $S_{gen}$ terms. All of the reversible entropy change terms disappear. The sum of the three irreversible terms is the total entropy change of the universe for the process that involves the three locations. The sum must be positive. It is indeed positive in this case because the restriction on any one $S_{gen}$ is that it must be defined as positive when an irreversible process occurs.
Summary
The sign of the entropy change for a process that occurs in a system or in the surroundings is not a metric of whether the process is or is not "possible". It is a sign of the direction of heat flow in or out. It is a sign of whether the system has an increase or a decrease in order.
The sign of the total entropy change of the universe is not a metric of whether a process is or is not "possible". The technical statement is instead precisely that total entropy change is a metric of whether the process is spontaneous as it is proposed.
Entropy change of a defined control volume includes both reversible and irreversible values. The former is $\Delta S_{rev} = \int \delta q_{rev}/T$. The latter is $\Delta S_{irr} \equiv S_{gen}$. The sign of $S_{gen}$ is always positive by definition.
Further Insights
We have an easier time to determine whether a process is or is not spontaneous when we use other thermodynamic state functions instead of entropy. For example, at constant temperature and pressure, we use the Gibbs energy $\Delta G$. We only need to consider the change of the system not of the universe, and the spontaneity criteria is $\Delta G_{sys,T,p} < 0$.
When a proposed process is not spontaneous as written, that does not mean it is not possible. We may have to force the process to go in the reverse direction. Electroplating is an example where we force the process by adding "other work" (electrochemical energy) to the system.
A: You don't specify your problem, so it is hard to answer. I will nevertheless suppose that the textbook (or whatever) dealt with everything that was required to correctly analyse the case. Therefore, if they haven't included any reference to the surroundingm, it means that the process is possible even if the system is isolated. Hence the transformation is possible as far as the second principle is concerned.
I am not sure to understand why you introduce 3 entropies either. Heat exchange between your system and the surroundings have effects on entropies of your system and the surroundings, and at the end only those two have to be taken into account.
[Edit after OP's editing of his question]
[Edit of the edit after discussion with OP in comment section]

Now if we inspect the total entropy change and it was positive, does this necessarily mean that S(gen A) is positive and S(gen B) is positive and S(gen C) is positive

No, it does not, of course. But this is not the point of the page you've scanned.
This page simply states that total entropy change is equal to total entropy generation (since exchange terms obviously cancel), hence is the sum of 3 positive terms, hence is positive.
It simply demonstrate that entropy generation being positive implies that so total entropy of an isolated system (the Universe as a whole being the isolated system considered here, as being the sum of the two control systems and the surroundings) must grow (since each constituant creates a positive or null entropy and exchange doesn't change the sum). 
I think you infered more from the book than actually written, which happens all the time! :)
A: For a reversible process you can have entropy change in the system and entropy change in the surroundings, but the total entropy change is zero. These changes are not called entropy generation. Entropy generation only occurs in an irreversible process. 
When we look at a process, we are looking at what is happening in the system. In an irreversible process of the system the entropy is generated in the system, not the surroundings. In order for the system to complete a cycle and have all its properties, including its entropy, return to the original equilibrium state, the system has to get rid of the entropy it generated as a result of the irreversibilities, whatever they may have been. It does this by transferring the entropy generated to the surroundings in the form of heat.  
In this way the total entropy change of the system is zero, but the entropy of the surroundings has increased due to the system dumping its excess entropy there. The total entropy change, system + surroundings is thus greater than zero, indicating the process was irreversible.
Hope this helps.
