A pension fund manager is considering investing in two shares A and B. It is est

Question

A pension fund manager is considering investing in two shares A and B. It is estimated that

(a)Share A will earn a dividend of 12% per annum and share B 4% per annum.

(b)Growth in the market value in one year of share A will be 10 ngwee per K1 invested and in B 40 ngwee per K1 invested.

He wants to invest the minimum total sum which will give dividend income of atleast K600 per annum and growth in one year of atleast K1, 000 on the initial investment. You are required to

(i)State the mathematical formulation of the problem

Compute the minimum sum to be invested to meet the manager’s objective by using the simplex method on the dual problem.  

Explanation / Answer

let X1 K be invested in A, X2 K in B Mathematical formulation X1 X2 RHS Equation form Minimize 1 1 Min X1 + X2 Constraint 1 0.12 0.04 >= 600 .12X1 + .04X2 >= 600 Constraint 2 0.1 0.4 >= 1000 .1X1 + .4X2 >= 1000 constraint 3 1 0 >= 0 X1 >= 0 constraint 4 0 1 >= 0 X2 >= 0 Solution 4545.46 1363.64 5909.09 Ans: 5909.09 K Calculations : Cj Basic Variables Quantity 1 X1 1 X2 0 artfcl 1 0 surplus 1 0 artfcl 2 0 surplus 2 0 artfcl 3 0 surplus 3 0 artfcl 4 0 surplus 4 Phase 1 - Iteration 1 1 artfcl 1 600 0.12 0.04 1 -1 0 0 0 0 0 0 1 artfcl 2 1,000 0.1 0.4 0 0 1 -1 0 0 0 0 1 artfcl 3 0 1 0 0 0 0 0 1 -1 0 0 1 artfcl 4 0 0 1 0 0 0 0 0 0 1 -1 zj 1,600 -1.22 -1.44 1 1 1 1 1 1 1 1 cj-zj 1.22 1.44 0 -1 0 -1 0 -1 0 -1 Iteration 2 1 artfcl 1 600 0 0.04 1 -1 0 0 -0.12 0.12 0 0 1 artfcl 2 1,000 0 0.4 0 0 1 -1 -0.1 0.1 0 0 0 X1 0 1 0 0 0 0 0 1 -1 0 0 1 artfcl 4 0 0 1 0 0 0 0 0 0 1 -1 zj 1,600 0 -1.44 1 1 1 1 2.22 -0.22 1 1 cj-zj 0 1.44 0 -1 0 -1 -1.22 0.22 0 -1 Iteration 3 1 artfcl 1 600 0 0 1 -1 0 0 -0.12 0.12 -0.04 0.04 1 artfcl 2 1,000 0 0 0 0 1 -1 -0.1 0.1 -0.4 0.4 0 X1 0 1 0 0 0 0 0 1 -1 0 0 0 X2 0 0 1 0 0 0 0 0 0 1 -1 zj 1,600 0 0 1 1 1 1 2.22 -0.22 2.44 -0.44 cj-zj 0 0 0 -1 0 -1 -1.22 0.22 -1.44 0.44 Iteration 4 1 artfcl 1 500 0 0 1 -1 -0.1 0.1 -0.11 0.11 0 0 0 surplus 4 2,500.00 0 0 0 0 2.5 -2.5 -0.25 0.25 -1 1 0 X1 0 1 0 0 0 0 0 1 -1 0 0 0 X2 2,500.00 0 1 0 0 2.5 -2.5 -0.25 0.25 0 0 zj 500 0 0 1 1 2.1 -0.1 2.11 -0.11 2 0 cj-zj 0 0 0 -1 -1.1 0.1 -1.11 0.11 -1 0 Iteration 5 0 surplus 3 4,545.45 0 0 9.0909 -9.0909 -0.9091 0.9091 -1 1 0 0 0 surplus 4 1,363.64 0 0 -2.2727 2.2727 2.7273 -2.7273 0 0 -1 1 0 X1 4,545.45 1 0 9.0909 -9.0909 -0.9091 0.9091 0 0 0 0 0 X2 1,363.64 0 1 -2.2727 2.2727 2.7273 -2.7273 0 0 0 0 zj 0 0 0 2 0 2 0 2 0 2 0 cj-zj 0 0 -1 0 -1 0 -1 0 -1 0 Phase 2 0 surplus 3 4,545.45 0 0 9.0909 -9.0909 -0.9091 0.9091 -1 1 0 0 0 surplus 4 1,363.64 0 0 -2.2727 2.2727 2.7273 -2.7273 0 0 -1 1 1 X1 4,545.45 1 0 9.0909 -9.0909 -0.9091 0.9091 0 0 0 0 1 X2 1,363.64 0 1 -2.2727 2.2727 2.7273 -2.7273 0 0 0 0 zj 5,909.09 1 1 -6.82 6.82 -1.82 1.82 0 0 0 0 cj-zj 0 0 6.8182 -6.8182 1.8182 -1.8182 0 0 0 0

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