donald waters, quantitative methods for business, 4 th edition © donald waters 2008 slide 12.1...

Donald Waters, Quantitative Methods for Business, 4th Edition © Donald Waters 2008

Slide 12.1

Chapter 12

Planning with linear programming


Slide 12.2

After finishing this chapter youshould be able to:

• Appreciate the concept of constrained optimisation• Describe the stages in solving a linear programme• Formulate linear programmes and understand the

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.


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.


Slide 12.4

There are three distinct stages in solving a linear programme:

• formulation – getting the problem in the right form

• solution – finding an optimal solution to the problem

• sensitivity analysis – seeing what happens when the problem is changed slightly.


Slide 12.5

Formulation contains

• 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.


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


Slide 12.7

• The feasible region is convex.

• The optimal solution is always at an extreme 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.


Slide 12.8

Sensitivity analysis finds what happens to the solution when

• Resources change

• The objective function changes


Slide 12.9

Changes to resources

• Here shadow prices show the value of each 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.


Slide 12.10

Changes to the objective function

• When the coefficients in the objective function 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.


Slide 12.11

Figure 12.1 Production problem for Growbig and Thrive


Slide 12.12

Figure 12.2 Graph of the blending constraint


Slide 12.13

Figure 12.3 Graph of the three constraints defining a feasible region


Slide 12.14

Figure 12.4 Profit lines for Growbig and Thrive


Slide 12.15

Figure 12.6 Moving the objective function line as far as possible away from the origin identifies the optimal solution


Slide 12.16

Figure 12.7 Identifying the optimal solution for worked example 12.4


Slide 12.17

Figure 12.8 Printout for the Growbig and Thrive problem


Slide 12.18

Figure 12.9 Using ‘Solver’ for a linear programme


Slide 12.19

Figure 12.10 Output from a LP package for worked example 12.6


Slide 12.20

Figure 12.11 Graph of solution for Amalgamated Engineering, worked example 12.6


Slide 12.21

Figure 12.12 Printout for West Coast Wood Products, worked example 12.7


Slide 12.22

Figure 12.13 Printout for problem 12.6

donald waters, quantitative methods for business, 4 th edition © donald waters 2008 slide 12.1...

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