AIMMS Optimization Modeling

model To illustrate an abstract model, the Teddy Bear Company is introduced. This company produces black and brown teddy bears in three sizes, and its owners consider the teddy bear in terms of an abstract model. That is, they describe everything they need to know about producing it: materials: fur cloth in black and brown, thread, buttons, different qualities of foam to stuff the bears, information on prices and suppliers of materials, three different sized patterns for cutting the fur, and assembly instructions for the three sizes of bears. In this abstract model of the teddy bear, there is just the basic information about the bear. A mathematical model can be used to derive further information. Chapter 1. Background 3 Mathematical models Suppose you want to know the cost of a small black teddy bear. Then you sum the material costs (material prices by the amounts used) and the production costs (assembly time by labor cost). The relationship between all these components can be expressed in general terms. Once formulated, you have developed a mathematical model. A mathematical model is an abstract model. It is a description of some part of the real world expressed in the language of mathematics. There may be some variables in the description—values that are unknown. If the model is formulated so that you can calculate these values (such as the cost of a small black teddy bear) then you can solve the model. Optimization models One class of mathematical models is referred to as optimization models. These are the main subject of this guide. For now, a small example will be given. Recall that the teddy bears are filled with foam which comes in small pieces and is available in different qualities. By choosing a certain mix you can achieve a specified “softness” for each size of teddy bear. The company wants to decide how much to buy of each quality. A constrained optimization model could be used to determine the cheapest mix of foams to yield a certain softness. Note that the optimization is in the determination of the cheapest combination (cost minimization), and that the constraint(s) are in terms of the softness requirement.