Category: Deterministic Decision Models

When dealing with simple linear regression, we typically use least squares to fit the coefficients of a simple linear model y = a + bx. Given a set of joint observations (xi, yi), i = 1, …, N, we define residuals and minimize the sum of squared residuals: This is actually a quadratic program, but because of the simplicity of constraints, we know from that…

The models we have described in the last section rely on two quite relevant limiting assumptions: Luckily, there is an array of modeling tricks that can be used to partially overcome these difficulties. In the next sections we illustrate a few of them, in order to show that the LP modeling framework is less restrictive…

Many real-life optimization problems relate with transportation of items on a network. This is clearly a relevant class of problems in supply chain management, but also many telecommunications problems involve networks on which data flow, rather than physical commodities. More surprisingly, some dynamic problems may be represented as network models on which items flow in…

In the production planning models that we have considered so far, there is a very precise way of producing each item type. When producing a car, you typically need an engine and four wheels. Factors cannot be substituted; there is no way to convince a customer to buy a car with 20 wheels and no…

In the previous two models for production planning there is a major omission: They do not involve any inventory buildup and depletion. From the familiar EOQ model, we know that there is one possible reason for building inventory, i.e., the presence of fixed ordering cost. A similar reason, which may be more relevant when producing…

The naive production mix model is just a starting point in modeling production planning, as many issues that make real-life models interesting and challenging are blatantly disregarded. We will proceed step by step, showing how more realistic features may be represented. In this section we consider one such issue, related to purchasing raw materials or…

Continuous linear programming (LP) problems are convex mathematical programs, for which extremely efficient solution methods are widely available. Therefore, real-life and large-scale problems can actually be tackled, provided that we are able to cast the decision problem in LP form. To squeeze a problem into the LP paradigm, we need the ability of formalizing decisions,…

Let us consider an abstract mathematical programming problem: Intuition would suggest that an unconstrained problem, where , is much easier to solve than a constrained one. Moreover, the same intuition would suggest that the larger the problem, in terms of the number of decision variables and constraints, the more difficult is solving it. In fact, this…

A mathematical programming problem is called a linear programming (LP) problem when all the constraints and the objective function are expressed by linear affine functions, as in the following case: An LP model can involve inequality and equality constraints, as well as simple bounds. The general form of a linear programming problem is where we denote the…

In Part I we got acquainted with two elementary and prototypical optimization models, which we recall here for readers’ convenience: Looking at these two examples, we notice similarities and differences: In general and abstract terms, we may refer to an optimization problem in the following form where S, the feasible set, is a subset of . If , we have…