Mathematical Programming

Mathematical programming uses mathematical variables and expressions to model problems. In the formulation step of the Operations Research (OR) methodology we identify the key quantifiable decisions, restrictions and goals from the problem description, and capture their interdependencies in a mathematical programming model also known as a mathematical programme.

Formulating a Mathematical Programme

We can break the formulation process into 4 key steps:

1. Identify the Decision Variables paying particular attention to units (for example: we need to decide how many hours per week each process will run for).
2. Formulate the Objective Function using the decision variables, we can construct a minimise or maximise objective function. The objective function typically reflects the total cost, or total profit, for a given value of the decision variables.
3. Formulate the Constraints, either logical (for example, we cannot work for a negative number of hours), or explicit to the problem description. Again, the constraints are expressed in terms of the decision variables.
4. Identify the Data needed for the objective function and constraints. To solve your mathematical programme you will need to have some "hard numbers" as variable bounds and/or variable coefficients in your objective function and/or constraints.

Solving a Mathematical Programme

For relatively simple or well understood problems the mathematical programme can often be solved to optimality (i.e., the best possible solution is identified) using algorithms such as the Revised Simplex Method, interior point methods, or branch-and-bound. However, some industrial problems would take too long to solve to optimality using these classical optimisation techniques. Often these problems are solved using heuristic methods (such as Tabu search and Simulated Annealing) which do not guarantee optimality. The best solution method for a mathematical programme is highly dependent on the type of mathematical programme being solved.

Types of Mathematical Programme

For more information about formulating and solving a mathematical programme see the topics for the specific types of mathematical programme:

• Linear Programming
• Integer Programming
  • Set Partitioning, Packing and Covering
• Nonlinear Programming
• Network Optimisation

-- TWikiAdminGroup - 20 Feb 2008