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Tutorial 2

# Tutorial 2 - 2 x ≥ 2 x 1 x 1 1 2 3 4 5 6 7 8 2 3 4 5 6 7...

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IELM201 Tutorial 2 1/6/11 1. Example: Production time per batch (hrs) Time available per week (hrs) Plant Product 1 Product 2 1 1 0 4 2 0 2 12 3 3 2 18 Profit: \$3K \$5K Let 1 x = number of units of product 1 produced 2 x = number of units of product 2 produced maximize 3 1 x + 5 2 x subject to 1 x 4 2 2 x 12 3 1 x + 2 2 x 18 1 x , 2 x 0 1 x 2 x 1 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8

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IELM201 Tutorial 2 1/6/11 maximize 3 1 x + 5 2 x subject to 1 x 4 2 2 x 12 3 1 x + 2 2 x 18 1 x , 2 x 0 The Optimal Solution: Observations: The optimal solution cannot fall in the interior of the feasible region An optimal solution must occur at a “corner point” of the feasible region 2 x 1 x 1 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8
IELM201 Tutorial 2 1/6/11 Special Situations 1. Optimal Solution not unique: Example: maximize 7.5 1 x + 5 2 x New objective subject to 1 x 4 2 2 x 12 3 1 x + 2 2 x 18 1 x , 2 x 0 1 x 2 x 1 x 1 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8

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IELM201 Tutorial 2 1/6/11 2. No feasible solution: Example: maximize 3 1 x + 5 2 x subject to 1 x 4 2 2 x 12 3 1 x + 2 2 x 18 1 x + 2 x 9 Additional constraint 1 x ,

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Unformatted text preview: 2 x ≥ 2 x 1 x 1 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 IELM201 Tutorial 2 1/6/11 3. Objective function is unbounded : Example: maximize 3 1 x + 5 2 x subject to 1 x ≥ 4 New direction of inequality 2 2 x ≤ 12 3 1 x + 2 2 x ≥ 18 New direction of inequality 1 x , 2 x ≥ 2 x 1 x 1 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 IELM201 Tutorial 2 1/6/11 2. Consider the following LP model. Identify the optimal solution on the graph. minimize Z=2 1 x + 2 x subject to-1 x +2 2 x ≤ 10 1 x- 2 2 x ≤ 4 1 x + 2 x ≥ 8 1 x + 2 2 x ≥ 20 1 x , 2 x ≥ This example has a unique optimal solution with unbounded feasible region. ( 1 x , 2 x )=(5, 7.5) and Z=17.5...
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