"Would you expect the objective function of an integrated refinery model (two or more sites) to normally be lower or higher than the sum of the value of the individual models?", ask some readers who have been debating this point with their colleagues.  
If there are no connections between the sites, then the solution to the combined model should be the sum of the two individual ones (ignoring the possibility of variation due to alternative local solutions.)   This is usually the first test case you run when combining models together, to prove that they are compatible and you haven’t broken anything.   However, the point of running an integrated model is to try to take advantage of synergies between sites or to represent overall constraints, so you will certainly continue on to add some additional data to the integrated model.   Whether the overall solution ends up being greater or less than the sum of the parts depends on whether the opportunities outweigh the limitations and the extent to which they are represented in the individual models.     
Suppose that site A is tight on a product property, while site B has giveaway.  Some component could be moved from B to A, allowing A to make more product and therefore more money – provided the product can be sold; B might make less product, having lost some material, but if it can be replaced in the blending mix, or was being used in a lower value item, then you make more money overall when the component is moved.   So the joint model could profit from this option.   This component transfer might already be represented in the individual models as an output from B and an input to A.  If so, some of the potential value is already taken into account, reducing the additional profit to be found in the integrated model.  However, when the two models are not connected, the component has to be represented as a fixed quality stream.   Only when the models are connected can the solutions for the individual sites adapt in response to the overall profitability – adjusting the quality at B, to create the best transfer to A, and the blending at A to be placed to make best use of what B can send.   So I would expect the integrated model to take better advantage of a synergy than the individual models and find a better value than the sum of the parts.
There are also, though, overall constraints to be taken into account.   Suppose that both individual models are presented with a set of available feedstocks and the same ones turn up in the optimal solutions for A and B.  If the total required is more than what is available overall --- if both plans include a cargo of a particular crude but you can only buy one in the time frame -- then the individual models have over-optimized.   An integrated model could limit the total of this crude across the two sites - - or even allocate the whole cargo to A or B if the shipment can't be split between them.  It would generate a lower value solution – but it would be a more realistic plan. 
Double counting might also occur in your product sales in the stand-alone models.  If there are customers that could be supplied from either site, how do you represent the demands in each model?   If you simply allocate a share to each, you are not optimizing, so an integrated model might find a better solution.   If you put the full amount in each model, as if they were independent decisions, you are assuming more product than the market requires.  An integrated model with a single overall limit would predict less profit, but be more realistic.  
An Answer
So, in conclusion, integrated models might find solutions with more or less value than the sum of the separate models depending on whether synergies or constraints dominate and how these same issues were handled in the individual models. The usefulness of the combined model should not be judged on the change in solution value, but on the degree to which it is able to identify useful opportunities and suggest more realistic plans.
Thanks to Philip Henke and Lukas Rautenbacher for the question.

From Kathy's Desk, 13 March 2018.

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