Find the feasible region. Do this just as you would for a linear programming problem. Use the constraints of the problem to determine the limited region in which solutions may exist. A convenient way to do this is through graphing. For example, if your constraints are x + 3y <= 7, 2x + y <=7 and the knowledge that "x" and "y" cannot be negative, then you can draw these inequalities on a graph. The region should appear as a quadrilateral.
List all of the integer solutions in the feasible region. Unlike linear programming problems, integer programming problems yield a countable set of solutions, which you can list after observing the feasible region. If you are using graph paper, the possible integer solutions will be immediately visible. If you are graphing via other means, place a dot at each integer ordered pair inside the feasible region.
Write the list of possible solutions as an ordered pair. For the feasible region given in the example, the possible integer solutions are (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) and (2, 0).
Find the integer solution that maximizes -- or minimizes, depending on how the problem is stated -- the objective function. Plug the possible solutions into the objective function and compare the results. The result that is highest or lowest is the maximum or minimum possible value for the objective function. Thus, the solution to the linear programming problem is the integer ordered pair that gives this optimal value. For the example, if you are given the objective function "maximize x + y" you can easily see that the point (1, 2) in the set of possible solutions satisfies the objective function with a value of 3, the highest possible value. Therefore, for this example the solution is (1, 2).