Page 99 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes 4.5 Keywords

Dual Problem: In the dual problem, the dual vector multiplies the constants that determine the
positions of the constraints in the primal.

Duality principle: In optimization theory, the duality principle states that optimization problems
may be viewed from either of two perspectives, the primal problem or the dual problem.
Primal Problems: In the primal problem, the objective function is a linear combination of n
variables.

4.6 Review Questions

Determine the duals of the given problems.
1. Minimise Z = 12x + 26x + 80x
1 2 3
2x + 6x + 5x  4
1 2 3
4x + 2x + x  10
1 2 3
x + x + 2x  6
1 2 3
With all variables non-negative.
2. Minimise Z = 3x + 2x + x + 2x + 3x
1 2 3 4 5
Subject to: 2x + 5x + x + x  6
1 2 4 5
4x – 2x + 2x + 3x  5
2 3 4 5
x – 6x + 3x + 7x +5x  7
1 2 2 4 5
With all variables non-negative.
3. Maximise Z = 6x – x + 3x
1 2 3
Subject to 7x + 11x + 3x  25
1 2 3
2x + 8x + 6x  30
1 2 3
6x + x + 7x  35
1 2 3
With all variables non-negative.

4. Minimise Z = 10x + 15x + 20x + 25x
1 2 3 4
Subject to: 8x + 6x – x + x  16
1 2 3 4
3x + 2x – x  20
1 3 4
With all variables non-negative.
5. Minimise Z = x + 2x + x
1 2 3
Subject to x + x = 1
2 3
3x + x + 3x = 4
1 2 3
With all variables non-negative.

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