Slide 12.1
Chapter 12
Planning with linear programming
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.2
After finishing this chapter you
should be able to:
• Appreciate the concept of constrained optimisation
• Describe the stages in solving a linear programme
• Formulate linear programmes and understand the
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assumptions
Use graphs to solve linear programmes with two
variables
Calculate marginal values for resources
Calculate the effect of changing an objective
function
Interpret printouts from computer packages.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.3
Linear programming is a way of solving some
problems of constrained optimisation
• Constrained optimisation has:
– an aim of optimising – either maximising or
minimising – some objective.
– a set of constraints that limit the possible
solutions.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.4
There are three distinct stages in solving
a linear programme:
• formulation – getting the problem in the right
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•
form
solution – finding an optimal solution to the
problem
sensitivity analysis – seeing what happens when
the problem is changed slightly.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.5
Formulation contains
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decision variables
an objective function
a set of constraints
a non-negativity constraint.
• Formulating the problem is generally the most
difficult part, as it can need considerable skills.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.6
Finding a solution needs a lot of
repetitive arithmetic
• This is always done by computer.
• We can illustrate the general approach with a
graph for two variables.
Figure 12.5
Superimposing the objective function on the feasible region
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.7
• The feasible region is convex.
• The optimal solution is always at an extreme
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point of the feasible region.
The objective function identifies the optimal
point.
At the optimal solution some constraints are
limiting, and others have a slack.
Formal procedures to find optimal solutions are
based on the simplex method.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.8
Sensitivity analysis finds what happens
to the solution when
• Resources change
• The objective function changes
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.9
Changes to resources
• Here shadow prices show the value of each
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additional unit of resource.
Then each additional unit of resource increases
the objective function by the shadow price.
For small changes the optimal solution remains
at the same extreme point.
Shadow prices are only valid within certain limits
before the optimal solution moves to another
extreme point.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.10
Changes to the objective function
• When the coefficients in the objective function
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change, its gradient changes.
We can calculate the effects of these on the
optimal solution.
For small changes, the optimal solution remains
at the same extreme point.
For larger changes, the optimal solution moves
to another extreme point.
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.11
Figure 12.1
Production problem for Growbig and Thrive
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.12
Figure 12.2
Graph of the blending constraint
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.13
Figure 12.3
Graph of the three constraints defining a feasible region
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.14
Figure 12.4
Profit lines for Growbig and Thrive
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.15
Moving the objective function line as far as possible away from the origin
identifies the optimal solution
Figure 12.6
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.16
Figure 12.7
Identifying the optimal solution for worked example 12.4
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.17
Figure 12.8
Printout for the Growbig and Thrive problem
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.18
Figure 12.9
Using ‘Solver’ for a linear programme
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.19
Figure 12.10
Output from a LP package for worked example 12.6
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.20
Figure 12.11
Graph of solution for Amalgamated Engineering, worked example 12.6
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.21
Figure 12.12
Printout for West Coast Wood Products, worked example 12.7
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008
Slide 12.22
Figure 12.13
Printout for problem 12.6
Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008