Graphical_Solution_Method_for_LP

# Graphical_Solution_Method_for_LP - 1 Linear Programming...

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1 Graphical Solution Method for  Linear Programming Model

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2 Linear Programming Model of Catch-Big Problem Max 500,000x1 + 700,000x2 s.t. 20,000 x1 + 30,000 x2 < 190,000 x1 < 6 x1 + x2 < 8 x1, x2 > 0
3 Feasible Solutions and Optimal Solution § Feasible solution : the values of decision variables that satisfy all the constraints § Feasible region : all the feasible solutions § Optimal solution : feasible solution with the largest (or smallest) objective function value for maximization (or minimization)

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4 1 2 3 4 5 6 § Values for decision variables points in a plane with axes: Picture the Values For Decision Variables 5 4 3 2 1 x 2 x1 0
5 Example 1: LP Formulation § A simplification of the linear programming model: Max 5x1 + 7x2 (in 100,000) s.t. 2x1 + 3x2 < 19 (in 10,000) x1 < 6 x1 + x2 < 8 x1, x2 > 0

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6 Example 1: Graph Constraints § Graph the region of points satisfying 2x1 + 3x2 < 19 : 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x 2 x1
7 Example 1: Graph Constraints § Graph the region of points satisfying x1 < 6 : 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x 2 x1

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8 Example 1: Graph Constraints § Graph the region of points satisfying x1 + x2 < 8 : 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x 2 x1
9 Example 1: Graph Constraints § Graph the region of points satisfying all constraints : 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x 2 x1

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10 x2 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 § The region of points that satisfy all constraints : Example 1: Feasible Region x1 Feasible Region
11 Example 1: How Did We Graph Constraints? Constraint 1 : Set x2 = 0, then x1 = 9.5. Set x1 = 0, then x2 = 6.33. Connect (9.5, 0) and (0, 6.33). Using a verification point (0, 0), we see that the “<’’ side is to its left. Constraint 2

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Graphical_Solution_Method_for_LP - 1 Linear Programming...

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