Month: February 2023

The knapsack model and its variants are pure binary programming models. In this section we get acquainted with a quite common mixed-integer model, arising when the cost structure related to an activity cannot be represented in simple linear terms. The fixed-charge problem is one such case. Let decision variable x ≥ 0 represent the level of an activity. The… Continue reading Fixed-charge problem and semicontinuous decision variables

Let us consider a trivial model for capital budgeting decisions. We must allocate a given budget B of money to a set of N potential investments. For each investment opportunity, we know We would like to select the subset of investments that yields the highest total profit, subject to a limited budget B. This looks like a portfolio optimization model;… Continue reading Knapsack problem

As we have already pointed out, integer programming models may pop up when there is a need to restrict purchase or production decisions to integer quantities, maybe multiples of a standard batch. However, the most common reason for using such models is by far the inclusion of logical decisions. In this section we use a set… Continue reading BUILDING INTEGER PROGRAMMING MODELS

Sometimes, we face management problems with quite complicated constraints, which seem to defy the best modeling efforts. Column-based model formulations are a formidable tool, which is again best illustrated by a simple example, namely, a stylized staffing problem. Imagine that we are running a post office, or something like that, with a lot of counters;… Continue reading Column-based model formulations

An optimization model need not have a unique optimal solution. As we have pointed out in Section 12.1.1, the following can occur: Commercial solvers are able to spot infeasible mathematical programs, but, from a practical perspective, we cannot just report that, leaving the decision maker without a clue. It would be nice to provide her with… Continue reading Elastic model formulations

Goal programming is one way of dealing with conflicting objectives, but it requires the assessment of weights and targets. Unfortunately, it may be very difficult, or even unethical, to figure out weights. As an example, consider the tradeoff between the cost of a production process and its pollution level. Sometimes, we would like to visualize… Continue reading Multiobjective optimization

The deviation variables that we have utilized in order to formulate alternative regression models as LPs have other uses as well. Let us consider a generic optimization problem over a feasible set S. A standard complication of real-life decision problems is that there is not just one criterion to evaluate the quality of a solution, but… Continue reading Goal programming

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… Continue reading Alternative regression models

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… Continue reading  A REPERTOIRE OF MODEL FORMULATION TRICKS

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… Continue reading Network optimization