Given the following all-integer linear program: questions 21-24 use the program from 20

Given the following all-integer linear program:

Max	3x1 + 2x2
s.t.	3x1 + x2 9
	x1 + 3x2 7
	x1 + x2 1
	x1, x2 0 and integer

Solve the problem as a linear program ignoring the integer constraints. Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂.

21What solution is obtained by rounding fractions greater than or equal to 1/2 to the next larger number? Show that this solution is not a feasible solution. By rounding x₁ and x₂ to the nearest integer, we get x₁ =___ and x₂ =____ .This point is ___ (feasible / infeasible).

22What is the solution obtained by rounding down all fractions? Is it feasible? By rounding x₁ and x₂ down, we get x₁ =___ and x₂ =_____This point is____ (feasible / infeasible). If feasible, what is the objective value = _____. (If it is not feasible, simply write 0 for the obj value.)

23Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers, and show that the feasible solution obtained in the previous part is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

A partially filled table is provided for your convenience. Given the possible x₁ values, complete the missing values for x₂ and objective values.

Alternative	x1	x2	z
1	0
2	1
3	2
4	3
5	0
6	1
7	2
8	1

24Run the model in Excel Solver, this time adding integrality condition as well. (Make sure all decision variables can only take integer values)What is the optimal solution and its objective value? (Note: you can double-check your answer from the table you filled in the previous part)

x1
x2
Objective

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