(From Leslie Chandrakantha, Example 1) A trust office at the Blacksburg National

Question

(From Leslie Chandrakantha, Example 1) A trust office at the Blacksburg National Bank needs to determine how to invest S100,000 in following collection of bonds to maximize the annual return Maturity Risk Tax-Free Annual Return 95% 8.0% 9.0% 9.0% 9.0% Bond Long Short Long Long Short High Low Low High High Yes Yes No Yes No The officer wants to invest at least 50% of the money in short term issues and no more than 50% in high-risk issues. At least 30% of the funds should go in tax-free investments, and at least 40% of the total return should be tax free. Formulate a LP model for this problem (2 pts)

Explanation / Answer

Let amounts to be invested in bonds A, B,C D and E are X1, X2, X3, X4 and X5 respectively.

The corresponding annual returns accordingly will be :

Bond A : 0.095.X1

Bond B : 0.08.X2

Bond C : 0.09.X3

Bond D : 0.09.X4

Bond E : 0.09.X5

Thus the objective function will be ( To maximize cumulative annual returns ):

Maximize Z = 0.095.X1 + 0.08.X2 + 0.09.X3 + 0.09.X4 + 0.09.X5

Given, X1, X2, X3.X4,X5 > = 0

Since the officer wants to invest at least 50% of the money in short term issues ( which are B and E ),

( X2 + X5) / ( X1 + X2 + X3 + X4 + X5 ) > = 0.5

Or, X2 + X5 > = 0.5.X1 + 0.5.X2 + 0.5.X3 + 0.5.X4 + 0.5.X5

Or, 0.5.X2 – 0.5.X1 – 0.5.X3 – 0.5.X4   + 0.5.X5 > = 0

Officer also wants to invest no more than 50% in high risk issues ( which are A, D and E ) ,

( X1 + X4 + X5) / ( X1 + X2 +X3 + X4 + X5 ) < = 0.50,

Or, X1 + X4 + X5 < = 0.5.X1 + 0.5.X2 + 0.5.X3 + 0.5.X4 + 0.5.X5

Or, 0.5.X2 – 0.5.X1+ 0.5.X3 – 0.5.X4 – 0.5.X5 > = 0

Since at least 30% of funds should go into tax free investments ( which are A, B and D )

(X1 + X2 + X4 )/ ( X1 + X2 + X3 + X4 +X5 ) > = 0.3

X1 + X2 + X4 > = 0.3.X1 + 0.3.X2 + 0.3.X3 + 0.3.X4 + 0.3.X5

Or, 0.7.X1 + 0.7.X2 – 0.3.X3 + 0.7.X4 – 0.3.X5 > = 0

Since at least 40% of total return should be tax free :

( 0.095.X1 + 0.08.X2 + 0.09.X4) / ( 0.095.X1 + 0.08.X2 + 0.09.X3 +0.09.X4 + 0.09.X5) > = 0.40

Or, 0.095.X1 + 0.08.X2 + 0.09.X4 > = 0.038.X1 + 0.032.X2 + 0.036.X3 + 0.036.X4 + 0.036.X5

Or, 0.057.X1 + 0.048.X2 -0.036.X3 + 0.054.X4 – 0.036.X5 > = 0

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