Conceptualising Optimisation

It can be difficult to conceptualise exactly what mathematical modeling and optimisation is. The following simple example illustrates the power of optimisation.

The Situation

A firm must schedule production over a 10 week time horizon. Each good has a production cost of $45. If goods are produced in a given week a setup cost of $500 is incurred. Goods can be stored in inventory, a holding cost of $0.50 per unit per week applies. The firm's production plant has a maximum capacity of 145 units per week.

Find a Solution

Alter the production values in the blue shaded boxes to schedule production. Select 'Optimise' to view the cost minimising optimal solution. Were you able to obtain the optimal solution? Was your solution feasible?

Demand

Production

Inventory

The Model

The optimal solution is found by formulating and solving a mathematical model of the situation described above. The model expresses the situation as an objective function and a series of constraints. A solution is represented by a set of values for the production, inventory and setup variables. Optimisation algorithms are used to find the variable values that minimise the objective function while meeting the constraints.

Parameters in the problem are represented as follows:

The variables for the example situation are:

The objective is to minimise the total cost over all weeks. The total cost is the sum of variable cost, setup cost and holding cost. The objective function is stated below.

The first set of constraints ensure that inventory, production and demand balance across all weeks. One constraint exists for each week. These constraints are shown below.

The second set of constraints ensure that production is less than capacity in each week. They also turn on or off the setup cost using the binary variable 'Setup'.

Finally the setup variable is constrained to be binary and all other variables are constrained to be non-negative.