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

2. 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
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3. 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
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4. 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
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5. 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

6. 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
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7. 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
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8. Example– LP Formulation
(Objective function)
(Constraints)
(Nonnegativity constraints)
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9. 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
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Instructor Slides

10. Linear Programming Example

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

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

13. Example– Graphical LP: Step 1
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14. LP Example – Objective Function

15. LP Example – Objective Function

16. Example– Graphical LP: Step 2
Plotting constraints:
Begin by placing the nonnegativity constraints on a graph
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17. 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
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18. Example– Graphical LP: Step 2

19. Example– Graphical LP: Step 2

20. Example– Graphical LP: Step 2
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21. Example– Graphical LP: Step 2
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22. Example– Graphical LP: Step 2
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23. Storage Space Constraint
3A + 3B < 39 cubic feet

24. Storage Space Constraint

25. 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

26. Example– Graphical LP: Step 2
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27. Example– Graphical LP: Step 3
Feasible Solution Space
The set of points that satisfy all constraints simultaneously
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28. Feasible Solution Space

29. 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)
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30. Example– Graphical LP: Step 4
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31. LP Example – Objective Function

32. 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
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33. 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
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34. 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
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35. 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
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36. 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

37. 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
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38. Linear Programming Example

39. The Simplex Method Simplex method
A general purpose linear programming algorithm that can be used to solve problems having more than two decision variables
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40. 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
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41. Computer Solutions
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42. 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
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43. 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
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44. 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
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45. Computer Solutions
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46. 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
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47. Solver Results
Solver will incorporate the optimal values of the decision variables and the objective function into your original layout on your worksheets
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48. Answer Report
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49. Sensitivity Report
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50. 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
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51. 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
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52. 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
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53. 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
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54. 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
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