Given the Following All-Integer Linear Programming Problem A Solve the Problem Graphically as a Linear Program

Essay

Given the following all-integer linear programming problem:


$$\begin{array} { l } \operatorname { Max } \quad 3 x _ { 1 } + 10 x _ { 2 } \\\\\text { s.t. } 2 x _ { 1 } + x _ { 2 } \leq 5 \\x _ { 1 } + 6 x _ { 2 } \leq 9 \\\begin{aligned}x _ { 1 } - x _ { 2 } & \geq 2 \\x _ { 1 } , x _ { 2 } & \geq 0 \text { and integer }\end{aligned} \\\end{array}$$
a. Solve the problem graphically as a linear program.
b. Show that there is only one integer point and it is optimal.
c. Suppose the third constraint was changed to x₁ - x₂ > 2.1. What is the new optimal solution to the LP? ILP?