An LP problem with only two variables presents a simple case for which the solution can be derived using a graphical method. Before focusing on the steps involved in graphical method, we first take a look at a related theorem which roughly states that the optimal solution to an LP problem occurs at a corner point.
Theorem A.1 (The Fundamental Theorem of Linear Programming from [36]).
If the optimal value of the objective function in a Linear Programming problem exists, then that value (known as the optimal solution) must occur at one or more of the corner points of the feasible region.
Due to the scope of this thesis, no attention will be given to actually proving the above theorem and instead we then concentrate on the steps related to graphical method where we utilize the theorem.
Algorithm A.2 Graphical Method 1. Formulate the LP.
2. Represent constraints as equalities on x1, x2 coordinate plane and find the convex region
formed by them.
3. Plot the objective function
4. Find the vertices of the convex region and also the value of the objective function at each vertex. The vertex that gives the optimal value of the objective function gives the optimal solution to the problem.
It is important to be familiar with the terminology for solutions of the LP models. Solution is any specification of values for the decision variables; feasible solution is a solution for which all the constraints are satisfied; infeasible solution is a solution for which at least
one constraint is violated; feasible region is the collection of all feasible solutions; optimal solution is a feasible solution that has the most favorable value of the objective function; and most favorable value is the largest value is the objective function is to be maximized, whereas it is the smallest value if the objective function is to be minimized. There are in fact two basic methods to find the optimal solution.
1. If the problem is to find the point point in the feasible region, which maximizes the objective function we first draw the objective line when Z = 0 which passes through the origin. Then go on drawing lines parallel to this line till the line is farthest away from the origin and passes through only one point of the feasible region. In that case every point on that gives the maximum value of the objective function.
2. Here we determine all the corner points for the feasible region algebraically. Then eval- uate the objective function at each of the corner point and identify the optimal value of the objective function.
We now focus on some simple examples to understand the above concept better. Example A.1.
Solve the following LP problem using the graphical method. M aximize Z = 3x1+ 4x2
such that x1+ x2 ≤ 450
2x1+ x2 ≤ 600
x1, x2 ≥ 0
Evaluating the objective function at each of the corner points:
A ≡ (0, 450), ZA= 0 + 4× 450 = 1800
B ≡ (150, 300), ZB = 3× 150 + 4 × 300 = 1650
C ≡ (300, 0), ZC = 3× 300 + 0 = 900
0 x1(×100) x2(×100) -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 6 7 x 1+ x 2 ≤ 450 2 x 1 + x 2 ≤ 600 Z = 0 Z = 600 A C B O i
Figure A.1: Feasible region corresponding to Example A.1
Optimal solution occurs atA, where the maximum value is Z = 1800 at x1 = 0, x2 = 450.
Example A.2.
Solve the following LP problem using the graphical method. M inimize Z = 20x1+ 40x2
such that 36x1+ 6x2 ≥ 108
3x1+ 12x2 ≥ 36
20x1+ 10x2 ≥ 100
x1, x2 ≥ 0
Evaluating the objective function at each of the corner points: A≡ (0, 18), ZA= 0 + 40× 18 = 720
B ≡ (2, 6), ZB = 20× 2 + 40 × 6 = 280
C ≡ (4, 2), ZC = 20× 4 + 40 × 2 = 160
0 x1 x2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 36 x 1 + 10 x 2 ≥ 108 20 x 1 + x 2 ≥ 100 3x1 + 12x2≥ 36 A B C D i
Figure A.2: Feasible region corresponding to Example A.2
Optimal solution occurs at C, where the minimum value is Z = 160 at x1 = 4, x2 = 2.
However, there are three types of special cases of solutions to LPs. Multiple optimal solution is when function falls on more than one optimal point. If there are no points that simultaneously satisfy all constraints in the problem, we call it an infeasible problem. In some problems the feasible solution space formed by the constraints is not confined within a closed boundary. In these cases the objective function can sometimes increase indefinitely without ever reaching its maximum limit since it never reaches a constraint boundary. We call such a problem as an unbounded problem .
In this section we looked at a geometric method of solving an LP where we had only two unknowns and thus had a feasible region that is a subset of the real plane. As we introduce more variables to the LP, the graph of the feasible region becomes more complicated thus demanding another method to handle the LPs.