Neighborly Loan:

Jacob Marley, recently promoted to manager of the newest branch of the Neighborly Loan Company, is eager to please his new boss by loaning his annual $15 million budget profitably. Each local branch office at Neighborly generates profit by the interest income from three types of loans: first Mortgage loans on real estate at 7% annual interest, Automobile loans collateralized by liens on automobiles at 12% annual interest, and Signature loans with no collateral at 15% annual interest.  Having the highest risk, Signature loans carry the highest loan interest rate.
Neighborly's home office has set loan limits to guide branch managers and to protect the company from excessive amounts of high-risk loans. Neighborly requires each branch manager to place at least 60% of its loans into First Mortgages and no more than 10% of its loans in risky Signature loans.

If you make the mistake of expressing the mortgage and signatue constraints as percentage of budget rather than percentage of amount loaned, you get the right solutiion but an incorrect sensitivity analysis  The sensitivity analysis gives a dual price of 12 cents per dollar change in the amount budgeted, but actually changing the amount yields only an extra 9.3 cents on the dollar!.
Click here for spreadsheet

The reason is that changing the amount budgeted changes the maximum dollar amount for signature loans and the minimum dolar amount for first mortgages, which violates the "change just one constraint" assumption of the sensitivity analysis.



If you express the constraints as perecentages of the amount loaned, you get the correct solution and also the correct sensitivity analysis with the correct dual price of 9.3 cents per dollar change in amount budgeted.
Click here for spreadsheet

Note that the sensitivity report shows the RHS of the mortgage and signature loans equal to zero.
The constraints on signature loans and first mortgages are examples of a very important class of "proportionality constraints" in which a decision variable is constrained to be at least or at most some proportion of a total of which the variable itself is a part.  These will play a big role in "blending problems." 
In fact, Neighborly Loan can be viewed as a "blending problem" in which financial assets are blended into a portfolio.

These constraints are not expressed in standard form since there's a function on the right hand side.
Click here to see the problem expressed in standard form.
Note how much less human-friendly it is than the form on this and the previous page.
To Excel, the two are exactly the same.