McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Supplement 6 Linear Programming

6S-2 Linear Programming (LP) 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

6S-3 LP Models –Mathematical representations of constrained optimization problems –LP Model Components: Objective function –A mathematical statement of profit (or cost) for a given solution Decision variables –Amounts of either inputs or outputs Constraints –Limitations that restrict the available alternatives Parameters –Numerical constants

6S-4 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

6S-5 Model Formulation 1.List and define the decision variables (D.V.) –These typically represent quantities 2.State the objective function (O.F.) –It includes every D.V. in the model and its contribution to profit (or cost) 3.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 4.Non-negativity constraints

6S-6 Example– LP Formulation (Objective function) (Constraints) (Nonnegativity constraints)

6S-7 Graphical LP –A method for finding optimal solutions to two-variable problems –Procedure 1.Set up the objective function and the constraints in mathematical format 2.Plot the constraints 3.Indentify the feasible solution space –The set of all feasible combinations of decision variables as defined by the constraints 4.Plot the objective function 5.Determine the optimal solution

6S-8 Example– Graphical LP: Step 1

6S-9 Example– Graphical LP: Step 2 Plotting constraints: –Begin by placing the nonnegativity constraints on a graph

6S-10 Example– Graphical LP: Step 2 Plotting constraints: –Other constraints 1.Replace the inequality sign with an equal sign. 2.Determine where the line intersects each axis 3.Mark these intersection on the axes, and connect them with a straight line 4.Indicate by shading, whether the inequality is greater than or less than 5.Repeat steps 1 – 4 for each constraint

6S-11 Example– Graphical LP: Step 2

6S-12 Example– Graphical LP: Step 2

6S-13 Example– Graphical LP: Step 2

6S-14 Example– Graphical LP: Step 2

6S-15 Example– Graphical LP: Step 3 Feasible Solution Space –The set of points that satisfy all constraints simultaneously

6S-16 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)

6S-17 Example– Graphical LP: Step 4

6S-18 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

6S-19 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 x 1 and x 2 where this occurs

6S-20 Redundant Constraints 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

6S-21 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

6S-22 Slack and Surplus Binding Constraint –If a constraint form 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

6S-23 The Simplex Method Simplex method –A general purpose linear programming algorithm that can be used to solve problems having more than two decision variables

6S-24 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 6S.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

6S-25 Computer Solutions

6S-26 Computer Solutions Click on Tools on the top of the worksheet, and in the drop-down 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

6S-27 Computer Solutions Add the 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

6S-28 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

6S-29 Computer Solutions

6S-30 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

6S-31 Solver Results Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets

6S-32 Answer Report

6S-33 Sensitivity Report

6S-34 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

6S-35 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

6S-36 Basic and Non-Basic Variables 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

6S-37 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

6S-38 Binding vs. Non-binding Constraints 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