The Graphical Method for Solving LP’s Investigación de Operaciones M. En C. Eduardo Bustos Farías 1 Key Terms in Graphing • • • • • Optimal solution Feasible solution space Corner point Redundant constraint Slack Investigación de Operaciones M. En C. Eduardo Bustos Farías 2 Formulating LP Models • Formulating linear programming models involves the following steps: 1. Define the decision variables. 2. Determine the objective function. 3. Identify the constraints. 4. Determine appropriate values for parameters and determine whether an upper limit, lower limit, or equality is called for. 5. Use this information to build a model. 6. Validate the model. Investigación de Operaciones M. En C. Eduardo Bustos Farías 3 Solving Linear Programming Problems • Graphical Technique ¾ First graph the constraints: the solution set of the system is that region (or set of ordered pairs), which satisfies ALL the constraints. This region is called the feasible set Investigación de Operaciones M. En C. Eduardo Bustos Farías 4 Solving Linear Programming Problems: Graphical Technique continued ■ Locate all the corner points of the graph: the coordinates of the corners will be determined algebraically It is important to note that the optima is obtained at the boundary of the solution set and furthermore at the corner points. For linear programs, it can be shown that the optima will always be obtained at corner points. Investigación de Operaciones M. En C. Eduardo Bustos Farías 5 Solving Linear Programming Problems: Graphical Technique continued ■ Determine the optimal value: test all the corner points to see which yields the optimum value for the objective function Objective function Feasible set Investigación de Operaciones M. En C. Eduardo Bustos Farías Optimum 6 Example Investigación de Operaciones M. En C. Eduardo Bustos Farías 7 Example x1 = quantity of server model 1 to produce x2 = quantity of server model 2 to produce maximize Z = 60x1+50x2 Subject to: Investigación de Operaciones M. En C. Eduardo Bustos Farías 8 Feasible Region Based on a Plot of the First Constraint (assembly time) and the Nonnegativity Constraint Investigación de Operaciones M. En C. Eduardo Bustos Farías 9 Completed Graph of the Server Problem Showing All of the Constraints and the Feasible Solution Space Investigación de Operaciones M. En C. Eduardo Bustos Farías 10 A Completed Graph of the Server Problem Showing the Assembly and Inspection Constraints and the Feasible Solution Space Investigación de Operaciones M. En C. Eduardo Bustos Farías 11 Finding the Optimal Solution • The extreme point approach – Involves finding the coordinates of each corner point that borders the feasible solution space and then determining which corner point provides the best value of the objective function. – The extreme point theorem – If a problem has an optimal solution at least one optimal solution will occur at a corner point of the feasible solution space. Investigación de Operaciones M. En C. Eduardo Bustos Farías 12 The The Extreme Extreme Point Point Approach Approach 1. Graph the problem and identify the feasible solution space. 2. Determine the values of the decision variables at each corner point of the feasible solution space. 3. Substitute the values of the decision variables at each corner point into the objective function to obtain its value at each corner point. 4. After all corner points have been evaluated in a similar fashion, select the one with the highest value of the objective function (for a maximization problem) or lowest value (for a minimization problem) as the optimal solution. Investigación de Operaciones M. En C. Eduardo Bustos Farías 13 Graph of Server Problem with Extreme Points of the Feasible Solution Space Indicated Investigación de Operaciones M. En C. Eduardo Bustos Farías 14 Extreme Point Solutions for the Server Problem Investigación de Operaciones M. En C. Eduardo Bustos Farías 15 The The Objective Objective Function Function (Iso-Profit (Iso-Profit Line) Line) Approach Approach • This approach directly identifies the optimal corner point, so only the coordinates of the optimal point need to be determined. – Accomplishes this by adding the objective function to the graph and then using it to determine which point is optimal. – Avoids the need to determine the coordinates of all of the corner points of the feasible solution space. Investigación de Operaciones M. En C. Eduardo Bustos Farías 16 The Server Problem with Profit Lines of $300, $600, and $900 Investigación de Operaciones M. En C. Eduardo Bustos Farías 17 Finding the Optimal Solution to the Server Problem Investigación de Operaciones M. En C. Eduardo Bustos Farías 18 Graphing—Objective Graphing—Objective Function Function Approach Approach 1. Graph the constraints. 2. Identify the feasible solution space. 3. Set the objective function equal to some amount that is divisible by each of the objective function coefficients. 4. After identifying the optimal point, determine which two constraints intersect there. 5. Substitute the values obtained in the previous step into the objective function to determine the value Investigación de objectiveM.function En C. Eduardo Bustos Faríasoptimum. 19 of the at the Operaciones A Comparison of Maximization and Minimization Problems Investigación de Operaciones M. En C. Eduardo Bustos Farías 20 Example Minimization Determine the values of decision variables x1 and x2 that will yield the minimum cost in the following problem. Solve using the objective function approach. Investigación de Operaciones M. En C. Eduardo Bustos Farías 21 Graphing the Feasible Region and Using the Objective Function to Find the Optimum for Example Investigación de Operaciones M. En C. Eduardo Bustos Farías 22 Example Example Investigación de Operaciones M. En C. Eduardo Bustos Farías 23 Summary of Extreme Point Analysis for Example 1 Investigación de Operaciones 1 M. En C. Eduardo Bustos Farías .57 24 Computing the Amount of Slack for the Optimal Solution to the Server Problem Investigación de Operaciones M. En C. Eduardo Bustos Farías 25 Special Conditions in LP Models Investigación de Operaciones M. En C. Eduardo Bustos Farías 26 Special Conditions in LP Models • A number of special conditions may occur in LP problems: – – – – Alternate Optimal Solutions Redundant Constraints Unbounded Solutions Infeasibility Investigación de Operaciones M. En C. Eduardo Bustos Farías 27 Example of Alternate Optimal Solutions X2 What if the price of Aqua-Spas generates a profit of $450 instead of $350? 250 objective function level curve 450X1 + 300X2 = 78300 200 MAX: 350X1 + 300X2 S.T.: 1X1 + 1X2 <= 200 150 9X1 + 6X2 <= 1566 12X1 + 16X2 <= 2880 100 X1 >= 0 X2 >= 0 alternate optimal solutions 50 0 0 Investigación de Operaciones 50 100 150 200 M. En C. Eduardo Bustos Farías 250 X1 28 Example of a Redundant Constraint X2 What if 225 pumps are available instead of 200? 1X1 + 1X2 <= 225 250 Pumps boundary line of tubing constraint 200 boundary line of pump constraint Redundant constraint 150 boundary line of labor constraint 100 Feasible Region 50 0 0 Investigación de Operaciones 50 100 150 200 M. En C. Eduardo Bustos Farías 250 X1 29 Example of an Unbounded Solution Consider the following problem: MAX: Z= X1 + X2 S.T.:1X1 + 1X2 >= 400 -1X1 + 2X2 <= 400 X1 >= 0 X2 >= 0 Investigación de Operaciones M. En C. Eduardo Bustos Farías 30 Example of an Unbounded Solution X2 1000 objective function X1 + X2 = 600 800 -X1 + 2X2 = 400 objective function X1 + X2 = 800 600 400 200 X1 + X2 = 400 0 0 Investigación de Operaciones 200 400 600 800 M. En C. Eduardo Bustos Farías 1000 X1 31 Example of Infeasibility Consider the following problem: MAX: Z= X1 + X2 S.T.:1X1 + 1X2 <= 150 1X1 + 1X2 >= 200 X1 >= 0 X2 >= 0 Investigación de Operaciones M. En C. Eduardo Bustos Farías 32 Example of Infeasibility X2 250 200 X1 + X2 = 200 feasible region for second constraint 150 100 feasible region for first constraint 50 X1 + X2 = 150 0 0 Investigación de Operaciones 50 100 150 200 M. En C. Eduardo Bustos Farías 250 X1 33 PROBLEMS. Hillier, Frederick S. & Hillier, Mark S. “Introduction to Management Science”. 2nd. Ed. USA, Mc Graw Hill - Irwin, 2003. 870 pp. M. En C. Eduardo Bustos Farías Investigación de Operaciones M. En C. Eduardo Bustos Farías 34 Investigación de Operaciones M. En C. Eduardo Bustos Farías 35 Investigación de Operaciones M. En C. Eduardo Bustos Farías 36 Investigación de Operaciones M. En C. Eduardo Bustos Farías 37 Investigación de Operaciones M. En C. Eduardo Bustos Farías 38 Investigación de Operaciones M. En C. Eduardo Bustos Farías 39 Investigación de Operaciones M. En C. Eduardo Bustos Farías 40 Investigación de Operaciones M. En C. Eduardo Bustos Farías 41 Investigación de Operaciones M. En C. Eduardo Bustos Farías 42 Investigación de Operaciones M. En C. Eduardo Bustos Farías 43