Introduction

We have covered tools to represent some standard forms of uncertainty. Our main aims were to understand the relationship between variables of interest and possibly to forecast their future values. Understanding how a system works is clearly essential in all scientific disciplines, including the social ones. However, in management there is a further step: moving from knowledge discovery to decision making. So far, we have just hinted at decision models every now and then. We move on to a systematic treatment of quantitative models and methods for decision making. In this first step, we disregard uncertainty and deal with deterministic problems. We merge decision models with probability and statistics to address the case of decision making under uncertainty. This will open up a world of challenging and rewarding models. Yet, we should always keep in mind that even the best decision model is always based on an approximate description of reality, and it should be regarded as a support tool, not a magical oracle. We outline a few complications arising in the practical world.

• Convex sets and convex/concave functions
• Local and global optimizers
• Quadratic forms and multivariable calculus

There, we covered unconstrained optimization of functions of one variable; here we deal with problems involving possibly many decision variables and constraints. Apart from a very few lucky cases, there is no hope of solving such problems analytically, and we must rely on numerical solution methods. However, here we place much more emphasis on model building than model solving. Extremely efficient and reliable software packages are commercially available to solve rather large models; so, it can be argued that only model building is relevant. Nevertheless, a modicum of familiarity with the underlying solution strategies is needed to choose the right solution method and to understand when and why a model is easy or difficult to solve. Furthermore, model building and solving are not always disconnected; sometimes, the proper formulation of a model may greatly improve the computational performance in solving it.

We begin with a classification of decision models in Section 12.1; we draw the line between linear and nonlinear programming models, as well as between convex and nonconvex optimization problems. This classification has a quite practical purpose, as it is related to solution methods available to solve each class of problems. As we shall see, some models can be tackled by surprisingly fast methods that can solve large-scale problems with a reasonable computational effort; other models are a much harder nut to crack, and we should understand why. After this, we turn to model building. A few prototypical linear programming models are described in Section 12.2. Then, in Section 12.3, we illustrate a few tricks of the trade that are helpful in coping with less standard cases. The full power of quantitative modeling is unleashed in Section 12.4, where we see how to represent quite intricate problems mathematically by using integer programming techniques, involving logical decision variables. Section 12.5 is a bit more theoretical and deals with nonlinear programming; still, in this section we introduce quite relevant concepts, such as shadow prices, which have an important economic and managerial significance. Finally, in Section 12.6, we get a glimpse of standard solution algorithms like the simplex method for continuous linear programming and the branch and bound method for integer linear programming. That section may be safely skipped, if the reader so wishes, since its content is not used anywhere in the remainder of the book; yet, we should also stress the fact that these approaches are widely available in commercial software packages, and having at least a rough idea of how they work may help in using them properly.