Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide 12.1 Chapter 12 Planning with linear programming

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

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

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

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

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

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.1 Production problem for Growbig and Thrive

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.2 Graph of the blending constraint

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.3 Graph of the three constraints defining a feasible region

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.4 Profit lines for Growbig and Thrive

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.6 Moving the objective function line as far as possible away from the origin identifies the optimal solution

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.7 Identifying the optimal solution for worked example 12.4

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.8 Printout for the Growbig and Thrive problem

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure 12.9 Using ‘Solver’ for a linear programme

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure Output from a LP package for worked example 12.6

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure Graph of solution for Amalgamated Engineering, worked example 12.6

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure Printout for West Coast Wood Products, worked example 12.7

Donald Waters, Quantitative Methods for Business, 4 th Edition © Donald Waters 2008 Slide Figure Printout for problem 12.6