Linear
Programming
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You should be able to:
LO 19.1 Describe the type of problem that would lend itself to solution
using linear programming
LO 19.2 Formulate a linear programming model from a description of a
problem
LO 19.3 Solve simple linear programming problems using the graphical
method
LO 19.4 Interpret computer solutions of linear programming problems
LO 19.5 Do sensitivity analysis on the solution of a linear programming
problem
19-2
 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
LO 19.1
19-3
1.
List and define the decision variables (D.V.)

2.
State the objective function (O.F.)

3.
It includes every D.V. in the model and its contribution to profit (or cost)
List the constraints



4.
These typically represent quantities
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
LO 19.2
19-4
 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. Identify the feasible solution space

4.
5.
LO 19.3
The set of all feasible combinations of decision variables as defined
by the constraints
Plot the objective function
Determine the optimal solution
19-5
LO 19.4
19-6
 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
LO 19.4
19-7
 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
LO 19.4
19-8
 For the non-negativity constraints, check the checkbox to
Make Unconstrained Variables Non-Negative
 Select Simplex LP as the Solving Method
 Click Solve
LO 19.4
19-9
LO 19.4
19-10
 Solver will incorporate the optimal values of the decision variables
and the objective function into your original layout on your
worksheets
LO 19.4
19-11
LO 19.4
19-12
LO 19.5
19-13
 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
LO 19.5
19-14
 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
LO 19.5
19-15