Final answer:
The student's task is to graph the constraints for an all-integer linear program and identify the feasible region, which is the intersecting area on a graph representing the solutions that meet all the given inequalities and are integers.
Step-by-step explanation:
The student's question involves graphing the constraints and the feasible region for an all-integer linear program with the objective function Maximize Z = 1x1 + 1x2, subject to certain inequalities. To graph the constraints, we plot each inequality on a coordinate axis with x1 on the horizontal axis and x2 on the vertical axis. The feasible region is where all these inequalities overlap and will be bounded by the most restrictive constraints.
To begin, you graph the lines representing the constraints:
- 5x1 + 7x2 ≤ 42
- 1x1 + 6x2 ≤ 18
- 2x1 + 1x2 ≤ 15
Plotting these on a graph: for each constraint, calculate the intercepts by setting x1 and then x2 to zero. After plotting these points, draw a line through them. Repeat for each constraint. Next, identify the area that satisfies all constraints; this is your feasible region. Remember, the feasible points must also be integers since the problem specifies an all-integer linear program.
Feasible region analysis is crucial as it defines the set of possible solutions that adhere to all constraints. In this case, the feasible region is a polygon on the graph where the integer solutions are located.