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Consider the following linear optimization model: (P) max s.t. x1 + 2x2 -x1 + x₂ ≤ 6 x12x₂ ≤ 4 x1, x2 > 0. 1. Determine all vertices and extreme rays of the feasible region of (P) by drawing its feasible region. 2. Using the extreme rays, argue that (P) is unbounded.

Consider the following linear optimization model: (P) max s.t. x1 + 2x2 -x1 + x₂ ≤ 6 x12x₂ ≤ 4 x1, x2 > 0. 1. Determine all vertices and extreme rays of the feasible region of (P) by drawing its feasible region. 2. Using the extreme rays, argue that (P) is unbounded.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following linear optimization model: (P) max s.t. x1 + 2x2 -x1 + x₂ ≤ 6 x12x₂ 4 x1, x2 > 0. 1. Determine all vertices and extreme rays of the feasible region of (P) by drawing its feasible region. 2. Using the extreme rays, argue that (P) is unbounded. 3. Put problem (P) in standard form. Find all basic solutions to this system; label them A, B, C, ... Indicate these solutions (using their labels) on the figure you drew in Part 1. 4. Use the simplex algorithm (starting from the basis composed of slack variables) to show that (P) is unbounded. When multiple variables are eligible to enter the basis, select the eligible variable with highest reduced cost. If multiple variables are eligible to leave the basis, select the eligible variable whose index is smallest. (When providing an answer to this problem, report (at least) the simplex dictionary obtained at each iteration, and state what variables are entering/leaving the basis.) 5. Represent the sequence of basic solutions you encountered during simplex on the figure you drew in Part 1.
Transcribed Image Text:Consider the following linear optimization model: (P) max s.t. x1 + 2x2 -x1 + x₂ ≤ 6 x12x₂ 4 x1, x2 > 0. 1. Determine all vertices and extreme rays of the feasible region of (P) by drawing its feasible region. 2. Using the extreme rays, argue that (P) is unbounded. 3. Put problem (P) in standard form. Find all basic solutions to this system; label them A, B, C, ... Indicate these solutions (using their labels) on the figure you drew in Part 1. 4. Use the simplex algorithm (starting from the basis composed of slack variables) to show that (P) is unbounded. When multiple variables are eligible to enter the basis, select the eligible variable with highest reduced cost. If multiple variables are eligible to leave the basis, select the eligible variable whose index is smallest. (When providing an answer to this problem, report (at least) the simplex dictionary obtained at each iteration, and state what variables are entering/leaving the basis.) 5. Represent the sequence of basic solutions you encountered during simplex on the figure you drew in Part 1.
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