Chapter 19 Linear Programming McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 19: Learning Objectives You should be able to: Describe the type of problem that would lend itself to solution using linear programming Formulate a linear programming model from a description of a problem Solve simple linear programming problems using the graphical method Interpret computer solutions of linear programming problems Do sensitivity analysis on the solution of a linear programming problem Instructor Slides 19-2

Linear Programming (LP) A powerful quantitative tool used by operations and other manages to obtain optimal solutions to problems that involve restrictions or limitations Applications include: Establishing locations for emergency equipment and personnel to minimize response time Developing optimal production schedules Developing financial plans Determining optimal diet plans Instructor Slides 19-3

LP Models LP Models Mathematical representations of constrained optimization problems LP Model Components: Objective function A mathematical statement of profit (or cost, etc.) for a given solution Decision variables Amounts of either inputs or outputs Constraints Limitations that restrict the available alternatives Parameters Numerical constants Instructor Slides 19-4

Linear Programming Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time Linear programming (LP) techniques consist of a sequence of steps that will lead to an optimal solution to problems, in cases where an optimum exists

LP Assumptions In order for LP models to be used effectively, certain assumptions must be satisfied: Linearity The impact of decision variables is linear in constraints and in the objective function Divisibility Noninteger values of decision variables are acceptable Certainty Values of parameters are known and constant Nonnegativity Negative values of decision variables are unacceptable Instructor Slides 19-6

Model Formulation List and define the decision variables (D.V.) These typically represent quantities State the objective function (O.F.) It includes every D.V. in the model and its contribution to profit (or cost) List the constraints Right hand side value Relationship symbol (≤, ≥, or =) Left Hand Side The variables subject to the constraint, and their coefficients that indicate how much of the RHS quantity one unit of the D.V. represents Non-negativity constraints Instructor Slides 19-7

Example– LP Formulation (Objective function) (Constraints) (Nonnegativity constraints) 19-8 Instructor Slides

Graphical LP Graphical LP A method for finding optimal solutions to two-variable problems Procedure Set up the objective function and the constraints in mathematical format Plot the constraints Indentify the feasible solution space The set of all feasible combinations of decision variables as defined by the constraints Plot the objective function Determine the optimal solution 19-9 Instructor Slides

Linear Programming Example

Linear Programming Example Find the quantity of each model to produce in order to maximize the profit

LP Example – Decision Variables A: # of model A product to be built B: # of model B product to be built

Example– Graphical LP: Step 1 Instructor Slides 19-13

LP Example – Objective Function

LP Example – Objective Function

Example– Graphical LP: Step 2 Plotting constraints: Begin by placing the nonnegativity constraints on a graph 19-16 Instructor Slides

Example– Graphical LP: Step 2 Plotting constraints: Replace the inequality sign with an equal sign. Determine where the line intersects each axis Mark these intersection on the axes, and connect them with a straight line Indicate by shading, whether the inequality is greater than or less than Repeat steps 1 – 4 for each constraint 19-17 Instructor Slides

Example– Graphical LP: Step 2

Example– Graphical LP: Step 2

Example– Graphical LP: Step 2 Instructor Slides 19-20

Example– Graphical LP: Step 2 Instructor Slides 19-21

Example– Graphical LP: Step 2 Instructor Slides 19-22

Storage Space Constraint 3A + 3B < 39 cubic feet

Storage Space Constraint

Linear Programming Formulation Objective - profit Maximize Z=60A + 50B Subject to Assembly 4A + 10B <= 100 hours Inspection 2A + 1B <= 22 hours Storage 3A + 3B <= 39 cubic feet A, B >= 0 Nonnegativity Condition

Example– Graphical LP: Step 2 Instructor Slides 19-26

Example– Graphical LP: Step 3 Feasible Solution Space The set of points that satisfy all constraints simultaneously Instructor Slides 19-27

Feasible Solution Space

Example– Graphical LP: Step 4 Plotting the objective function line This follows the same logic as plotting a constraint line There is no equal sign, so we simply set the objective function to some quantity (profit or cost) The profit line can now be interpreted as an isoprofit line Every point on this line represents a combination of the decision variables that result in the same profit (in this case, to the profit you selected) 19-29 Instructor Slides

Example– Graphical LP: Step 4 Instructor Slides 19-30

LP Example – Objective Function

Example– Graphical LP: Step 4 As we increase the value for the objective function: The isoprofit line moves further away from the origin The isoprofit lines are parallel Instructor Slides 19-32

Example– Graphical LP: Step 5 Where is the optimal solution? The optimal solution occurs at the furthest point (for a maximization problem) from the origin the isoprofit can be moved and still be touching the feasible solution space This optimum point will occur at the intersection of two constraints: Solve for the values of x1 and x2 where this occurs Instructor Slides 19-33

Redundant Constraints A constraint that does not form a unique boundary of the feasible solution space Test: A constraint is redundant if its removal does not alter the feasible solution space Instructor Slides 19-34

Solutions and Corner Points The solution to any problem will occur at one of the feasible solution space corner points Enumeration approach Determine the coordinates for each of the corner points of the feasible solution space Corner points occur at the intersections of constraints Substitute the coordinates of each corner point into the objective function The corner point with the maximum (or minimum, depending on the objective) value is optimal 19-35 Instructor Slides

Optimal Solution The intersection of inspection and storage Solve two equations with two unknowns 2A + 1B = 22 3A + 3B = 39 A = 9 B = 4 Z = $740

Slack and Surplus Binding Constraint Surplus Slack If a constraint forms the optimal corner point of the feasible solution space, it is binding It effectively limits the value of the objective function If the constraint could be relaxed, the objective function could be improved Surplus When the value of decision variables are substituted into a ≥ constraint the amount by which the resulting value exceeds the right-hand side value Slack When the values of decision variables are substituted into a ≤ constraint, the amount by which the resulting value is less than the right-hand side 19-37 Instructor Slides

Linear Programming Example

The Simplex Method Simplex method A general purpose linear programming algorithm that can be used to solve problems having more than two decision variables Instructor Slides 19-39

Computer Solutions MS Excel can be used to solve LP problems using its Solver routine Enter the problem into a worksheet Where there is a zero in Figure 19.15, a formula was entered Solver automatically places a value of zero after you input the formula You must designate the cells where you want the optimal values for the decision variables Instructor Slides 19-40

Computer Solutions Instructor Slides 19-41

Computer Solutions In Excel 2010, click on Tools on the top of the worksheet, and in that menu, click on Solver Begin by setting the Target Cell This is where you want the optimal objective function value to be recorded Highlight Max (if the objective is to maximize) The changing cells are the cells where the optimal values of the decision variables will appear Instructor Slides 19-42

Computer Solutions Add a constraint, by clicking add For each constraint, enter the cell that contains the left-hand side for the constraint Select the appropriate relationship sign (≤, ≥, or =) Enter the RHS value or click on the cell containing the value Repeat the process for each system constraint Instructor Slides 19-43

Computer Solutions For the nonnegativity constraints, enter the range of cells designated for the optimal values of the decision variables Click OK, rather than Add You will be returned to the Solver menu Click on Options In the Options menu, Click on Assume Linear Model Click OK; you will be returned to the solver menu Click Solve Instructor Slides 19-44

Computer Solutions 19-45 Instructor Slides

Solver Results The Solver Results menu will appear You will have one of two results A Solution In the Solver Results menu Reports box Highlight both Answer and Sensitivity Click OK An Error message Make corrections and click solve 19-46 Instructor Slides

Solver Results Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets 19-47 Instructor Slides

Answer Report 19-48 Instructor Slides

Sensitivity Report 19-49 Instructor Slides

Sensitivity Analysis Sensitivity Analysis Assessing the impact of potential changes to the numerical values of an LP model Three types of changes Objective function coefficients Right-hand values of constraints Constraint coefficients We will consider these 19-50 Instructor Slides

O.F. Coefficient Changes A change in the value of an O.F. coefficient can cause a change in the optimal solution of a problem Not every change will result in a changed solution Range of Optimality The range of O.F. coefficient values for which the optimal values of the decision variables will not change 19-51 Instructor Slides

Basic and Non-Basic Variables Decision variables whose optimal values are non-zero Non-basic variables Decision variables whose optimal values are zero Reduced cost Unless the non-basic variable’s coefficient increases by more than its reduced cost, it will continue to be non-basic Instructor Slides 19-52

RHS Value Changes Shadow price Amount by which the value of the objective function would change with a one-unit change in the RHS value of a constraint Range of feasibility Range of values for the RHS of a constraint over which the shadow price remains the same Instructor Slides 19-53

Binding vs. Non-binding Constraints have shadow price values that are equal to zero have slack (≤ constraint) or surplus (≥ constraint) Changing the RHS value of a non-binding constraint (over its range of feasibility) will have no effect on the optimal solution Binding constraint have shadow price values that are non-zero have no slack (≤ constraint) or surplus (≥ constraint) Changing the RHS value of a binding constraint will lead to a change in the optimal decision values and to a change in the value of the objective function Instructor Slides 19-54