This can be seen as a linear programming problem, mainly a network flow problem. 

Thank you for the suggestion. So, the solution need not be unique depending on the situation.

 Based on how you pose it it may not be unique. If you are familiar with optimization theory, if the gradient of the objective function is perpendicular with one of your constraints then you will have multiple optimal solution, atleast I know this is true for the linear programming case. I'm not sure if that is true with general non-linear optimization problems. 

The general theory is called the Symplex method. You have a certain resourse, that you want to distribute it in the best way possible between several resource needers.And you have certain equations as your restrictions. Or penalties in your case. In the simplest case i would suggest to read first ''Linear programming. New York: John Wiley & Sons Inc''.

Thank you

I will talk now as an applied mathematician and a specialist in analytical mathematical modelling who tries to understand where a mathematical problem comes from. I think that it is important to formulate the problem correctly. One can get correct solution for a wrong problem that will give non-applicable result. If the objective to maximize the quantity of alive fish in all cities, you do not need to transport - then there is no loss.

In your case I also have a doubt about formulation: 1 kg of fish dies per 1 km of transport. Suppose we have 1 ton of fish. Then 1kg dies the 1st km, … and the last kg dies on the 1000th km. If you carry only 10 kg of fish, then after 10 km all fish dies. I see no biological law behind that. It would be more natural to assume that probability to die is let say 1% per 1 km. Then for large number of fish (we use the law of large numbers) we will have exponential decline of alive stock  of fish with distance. This is similar to the process of decay of radioactive elements.

The second question is economical. There should be an opportunity to keep fish where it is caught – there will be no loss. Indeed, any point has its market price of fish. If in point A the price is 0.1$/kg, while in point B it is 1$/kg, it makes sense to transport it from A to B. But only if the sum of transport cost and the opportunity cost of dead fish (0.1 $/kg) is lower than the value of delivered fish. Also, will dead fish also have some value? (we almost never buy alive fish in shops)

Mathematically such reformulation will lead to some non-linearity, and thus direct differentiation of  objective function (one needs to think what it is - for example, economic profit) will give an optimal solution analytically.

Dear Yuri, Thank you for your valuable comments.

Actually, I could see the applications of a similar problem in neural network, power grid etc.

The original problem is as follows: We know that red blood cells are made in bone marrow and die after 120 days. I am trying to formulate the red blood cells movement from one place to another before they die. Another problem is the quantity of oxygen carried by the blood cells and their amount after x distance. Of course, the quantity is so large compared to the loss.

The fish problem is formulated in a simple way.

Apart from my real problem, if we think of power grid, then each city produces different amount of power in MWatt and they could be transferred between the cities so that the whole country has no power outage.

Instead of fish, if we think of plants, vegetables and grains, we can impose a similar criteria.

Dear Panchatcharam, so go along with correct problem formulation (for each application) and I wish you to obtain correct solution.

In the case of power transfer across cities I also would think in terms of energy loss in wires proportional to distance. You can see my model here: https://www.researchgate.net/publication/272176629_Competition_in_Electric_Energy_Network

I also tried to apply those ideas to water supply in a channel: https://www.researchgate.net/publication/228549496_Water_Spatial_Network_Pricing_45th_Congress_of_the_European_Regional_Science_Association

But these are more complex models in continuous space. You probably do not need this math but can borrow some ideas for problem formulation for your case.

Article Water: Spatial Network Pricing 45th Congress of the European...

Conference Paper Competition in Electric Energy Network *

Yeah there is, specifically you can use simplest method to obtain the solution for your problem and it's variant 

Dear Yuri and Jamilu,

Thank you for your suggestions