## (solution) I do not understand how the corner points (3,6) occurred! I

I do not understand how the corner points (3,6) occurred! I understand that when you solve by the elimination method you get the points (6,3)! Further, I understand how you get the points (0,0) and (0,6). However, graphically I think that I am doing something wrong because when I set 2X1 + 2X2 <=18 to zero for each variable I get the points 0/9 and 9/0. In essence, I am having trouble plotting the graph!

Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z Subject to = X1 + X2 2X1 + 2X2 &lt; 18 2X1 4X2 &lt; 0 3X2 &lt; 18 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (0, 0) 0 No (0, 6) 6 No (3, 6) 3 No (6, 3) ?3 Yes ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unique Optimal Page 1 Page 2 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z Subject to = 2X1 2X2 2X1 + X2 &lt; 4 3X1 + 3X2 &lt; 21 X1 2X2 &lt; 4 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (0, 0) 0 No (0, 4) ?8 No (1, 6) ?10 Yes (4, 0) 8 No (6, 1) 10 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unique Optimal Page 3 Page 4 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z Subject to = 100X1 X2 X1

X1 &lt; 6 2X2 &gt; 4 X2 &lt; 0 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (0, 2) ?2 No (2, 2) 198 No (6, 6) 594 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unbounded Page 5 Introduction to Linear Programming Page 6 A student has generated the following constraints for a linear programming problem. Minimize 8 .2 X 1 7 .0 X 2 6 .5 X 3 9 .0 X 4 Subject to : 6 X 1 2 X 2 5 X 3 7 X 4 820

X1

0.3

X1 X 2 X 3 X 4

X2 X3

0.2

X1 X 4

X 3 X1 X 4

All Decision Vaiables 0

However, she is confused about how to input the problem into EXCEL. Using the ?spreadsheet? below, indicate the

coefficients into the appropriate cells.

X1 X2 X3 X4 [ &lt;, =, &gt; ] RHS Minimize 8.2 7.0 6.5 9.0 Constraint #1 6.0 2.0 5.0 7.0 &gt; 820.0 Constraint #2 0.7 ?0.3 ?0.3 ?0.3 &gt; 0.0 Constraint #3 ?0.2 1.0 1.0 ?0.2 &lt; 0.0 Constraint #4 ?1.0 0.0 1.0 ?1.0 &gt; 0.0 Introduction to Linear Programming Page 7 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Maximize Z Subject to = 3X1 + 6X2 X1 + 2X2 &lt; 10 2X1 ? X2 &gt; 0 &lt; 6 X1 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (0, 0) 0 No (2, 4) 30 Yes (6, 0) 18 No (6, 2) 30 Yes ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Multiple Optimal Introduction to Linear Programming Page 8 Introduction to Linear Programming Page 9 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize

Subject to Z = X1 + X2 2X1 + 2X2 &lt; 14 ?3X1 + X2 &gt; 3 X1 ? 3X2 &gt; 3 Corner

Points All decision variables &gt; 0 Objective

Function

Value Optimal? ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

(0, 0) Introduction to Linear Programming Infeasible Page 10 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z = ?X1 + Subject to X2

X2 &lt; 8 2X1 + X2 &gt; 8 X1 ? X2 &gt; 1 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (3, 2) ?1 No (4, 0) ?4 No (9, 8) ?1 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unbounded Introduction to Linear Programming Page 11 Introduction to Linear Programming Page 12 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Maximize Z Subject to = X1 ? 200X2 3X1 + 9X2 &gt; 27 ?3X1 + X2 &lt; 3 2X2 &lt; 18 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (0, 3) ?600 No (2, 9) ?1,798 No (9, 0) 9 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unbounded Introduction to Linear Programming Page 13 Introduction to Linear Programming Page 14 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize

Subject to Z = X1 + X2 ?4X1 + 2X2 &gt; 8 2X1 ? 4X2 &gt; 8 3X1 + 3X2 &lt; 30 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Infeasible Introduction to Linear Programming Page 15 Introduction to Linear Programming Page 16 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z Subject to = X1 + X2 2X1 + 2X2 &lt; 20 X1 ? X2 &gt; 0 3X1 + 3X2 &gt; 12 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (2, 2) 4 Yes (4, 0) 4 Yes (5, 5) 10 No (10, 0) 10 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Multiple Optimal Solution Introduction to Linear Programming Page 17 Introduction to Linear Programming Page 18 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Maximize Z Subject to = X1 + 2X2 X1 + 3X2 &gt; 9 2X1 + X2 &gt; 8 X1 + X2 &lt; 10 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (0, 8) 16 No (0, 10) 20 Yes (3, 2) 7 No (9, 0) 9 No (10, 0) 10 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unique Optimal Introduction to Linear Programming Page 19 Introduction to Linear Programming Page 20 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z = ?X1 + Subject to X2

X2 &lt; 8 2X1 + X2 &gt; 8 X1 ? X2 &lt; 1 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (3, 2) ?1 Yes (0, 8) 8 No (9, 8) ?1 Yes ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Multiple Optimal Introduction to Linear Programming Page 21 Introduction to Linear Programming Page 22 Solve the following linear programming problem graphically (each line in the graphs is one unit). Indicate the feasible

region, list the corner points of the feasible region, determine the objective function value at the corner points, and indicate

the optimal solution (if any). Indicate the solution is ?Unique Optimal?, ?Multiple Optimal?, ?Infeasible?, or ?Unbounded? by

completing the box at the bottom of the problem.

Minimize Z Subject to = ?X1 +1000X2 3X1 ? X2 &gt; 0 X1 + X2 &gt; 4 X2 &lt; 6 All decision variables &gt; 0 (0, 0) Corner

Points Objective

Function

Value Optimal? (1, 3) 2,999 No (2, 6) 5,994 No (4, 0) ?4 No ?Unique Optimal? / ?Multiple Optimal? / ?Infeasible? /

?Unbounded?

Unbounded Introduction to Linear Programming Page 23 Do not write below this line Sample Exam #1G

24

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