Q. 1 Mr. Subhash has taken Rs. 100,000 from his father to invest them in a combination of only two stock portfolios with the maximum investment allowed in either portfolio set at Rs. 75,000. The first portfolio has an average return of 10% whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale from 0 to 10), and the second has 9. Since he wants to maximize his return, he will not accept an average rate return below 12% or a risk above 6. Hence, he then faces the important question. How much should he invest in each portfolio?
(a) Formulate the above as linear programming problem.
(b) Solve (a) by graphical method.
Q. 2 Consider the following LPP
(a) Solve using simplex method.
(b) Hence, using the sensitivity analysis, find the new optimal solution of the LPP if the availability of the second constraint is changed from 11 to 15.
Q. 3 Consider the following LPP
(a) Without using artificial variable(s), solve the given LPP (do not solve the the dual problem).
(b) Write the dual of the above problem.
Q.4 Consider the following LP
(a) Using M-method solve the above LP. Does the problem has alternative optimal solution? If so, find all the alternative optimal solutions.
(b) Write the dual of the above problem. Also, write the optimal solution of the dual problem (from the optimal table of part (a)).
(c) What can you conclude regarding the relationship of solution of primal and dual problems?
Q. 5 The following is an optimal LP tableau:
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Basic
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Solution
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1
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0
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0
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0
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3
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2
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?
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0
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0
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0
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1
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1
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-1
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2
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0
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0
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1
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0
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1
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0
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6
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0
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1
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0
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0
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-1
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1
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2
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The variablesx3,x4andx5are slacks in the original problem. Using matrix manipulations, reconstruct the original LP, and then compute the optimum objective value. Also, compute the optimum objective value by using dual objective function.