The Optimization Model in Algebraic Form

Consider the following simple linear programming problem. Our factory is building three products: TV sets, stereos and speakers. Each product is assembled from parts in inventory, and there are five types of parts: chassis, picture tubes, speaker cones, power supplies and electronics units. Our goal is to produce the mix of products that will maximize profits, given the inventory of parts on hand.

The Algebraic Form

The problem can be described in algebraic form as follows. The decision variables are the number of products of each type to build: x1 for TV sets, x2 for stereos and x3 for speakers. There is a fixed profit per unit for each product, so the objective function (the quantity we want to maximize) is:

Maximize 75 x1 + 50 x2 + 35 x3 (Profit)

Building each product requires a certain number of parts of each type. For example, TV sets and stereos each require one chassis, but speakers don't use one. The number of parts used depends on the mix of products built (the left hand side of each constraint), and we have a limited number of parts of each type on hand (the corresponding constraint right hand side):

Subject to:

1 x1 + 1 x2 + 0 x3 <= 450 (Chassis)

1 x1 + 0 x2 + 0 x3 <= 250 (Picture tubes)

2 x1 + 2 x2 + 1 x3 <= 800 (Speaker cones)

1 x1 + 1 x2 + 0 x3 <= 450 (Power supplies)

2 x1 + 1 x2 + 1 x3 <= 600 (Electronics)

Since the number of products built must be nonnegative, we also have the constraints x1, x2, x3 >= 0. Note that terms like 0 x3 are included purely to show the structure of the model - they can be either omitted or included in your data. Using this algebraic notation, we can start writing our RASON model.