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sensitivity_analysis_1 - USE OF MATRIX FORMULAS An Example...

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USE OF MATRIX FORMULAS An Example: Consider the LP and its standard form: max z = x 1 + 4 x 2 x 1 + 2 x 2 6 2 x 1 + x 2 8 x 1 0 x 2 0 max z = x 1 + 4 x 2 x 1 + 2 x 2 + s 1 = 6 2 x 1 + x 2 + s 2 = 8 x 1 0 x 2 0 s 1 0 s 2 0 QUESTION Given that the optimal basis is BV = { x 2 , s 2 } , find the optimal tableau. Saigal (U of M) IOE 310 1 / 20
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AN EXAMPLE First: Compute B - 1 . B = 2 0 1 1 , B - 1 = 1 2 0 - 1 2 1 , b = 6 8 . Second: Note that B - 1 Ax = B - 1 b gives 1 2 x 1 + x 2 + s 1 = 3 3 2 x 1 - 1 2 s 1 + s 2 = 5 Saigal (U of M) IOE 310 2 / 20
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Using c T BV = (4 , 0) and c T = (1 , 4 , 0 , 0) the objective row of the optimal tableau is: c T BV B - 1 A - c T which is obtained by first computing c T BV B - 1 = (2 , 0) and then c T BV B - 1 A - c T = (1 , 0 , 2 , 0) with the objective value: c T BV B - 1 b = 12 which gives the final tableau as z + x 1 +2 s 1 = 12 1 2 x 1 + x 2 + s 1 = 3 3 2 x 1 - 1 2 s 1 + s 2 = 5 Saigal (U of M) IOE 310 3 / 20
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SENSITIVITY ANALYSIS Changing the obj coeff of a nonbasic variable. Changing the obj coeff of a basic variable. Changing the RHS of a constraint. Changing the column of a non-basic variable. Adding a new activity. Adding a new constraint. Shadow Price. Saigal (U of M) IOE 310 4 / 20
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AN EXAMPLE I A furniture manufacturer has the following data: Desk Table Chair Available Lumber in Board ft 8 6 1 48 Finishing Hours 4 2 1.5 20 Carpentry Hours 2 1.5 0.5 8 Selling Price $60 $30 $20 Decision variables x 1 x 2 x 3 number produced Saigal (U of M) IOE 310 5 / 20
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AN EXAMPLE II The LP Model: max z = 60 x 1 + 30 x 2 + 20 x 3 8 x 1 + 6 x 2 + x 3 48 4 x 1 + 2 x 2 + 1 . 5 x 3 20 2 x 1 + 1 . 5 x 2 + 0 . 5 x 3 8 with x j 0 for each j = 1 , · · · , 3.
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