Mathematical modeling

By solving the previous two constraints equations, the values of x 1 and x 2 can be easilydetermined: the solution here is x 1 = 1 and x 2 = 7. The optimal decision is to produce1,000 notebooks and 7,000 desktops, for a profit of LE 7,750,000.2.3.4 Terminology of the Model SolutionThe final answer for the problem is always called solution, however in linearprogramming, any values for the decision variables ( x 1 , x 2 , …… ) that satisfies allconstraints are called a solution. So the following definitions are adopted: A feasiblesolution is a solution for which all the constraints are satisfied. An infeasible solution isa solution for which at least one constraint is violated.In the previous example, the points (2, 3) and (1, 6) are feasible solutions, while thepoints (4, 6) and ( -1, 3) are infeasible solutions. The feasible region is the collection ofall feasible solutions. In the example, the feasible region is the entire shaded area asshown in Fig. 2.6. It is possible for the problem to have no feasible solutions. This wouldhave happened in the example if it was required to have a net profit of at least LE 10,000per quarter. This constraint 750x 1 + 1000x 2 ≥ 10000 would eliminate the entire feasibleregion, so no mix of products would be able to produce this profit as shown in Fig. 2.7.(0, 10)(0, 8.3)(0, 7.5)(1, 7)optimumFeasibleregionActiveconstraintInactiveconstraintActiveconstraint(6.25, 0) (10, 0)(15, 0)Figure 2.6: Solution terminologySystems Analysis 20 Dr. Emad Elbeltagi