Tony Hürlimann

Mixed Integer Programming to Schedule a Single-Shift Workforce under Annualized Hours

Nowadays flexibility is a strategic concept for firms. Indeed workload has to follow, as close as possible, the development of demand throughout the year. However, firms cannot engage and dismiss employees according to production requirements. Thus, workforce scheduling becomes a delicate task. In this paper, four mixed integer programming models are developed to solve the workforce schedule problem for a single-shift. The annualized hour scenario is considered with respect to a set of Swiss legal constrains. Furthermore, the minimal required workforce is guaranteed and it is assumed that each employee is able to perform each task within the team. All employees are full-time workers.

LPL: A structured language for linear programming modeling

SummaryThis paper describes a specialized programming language, named LPL (Linear Programming Language), which may be used to build, modify and document LP models. The LPL language has been successfully applied by the authors to generate automatically MPSX input files of large LP models. The LPL Compiler translates LPL programs, which represent concise and readable LP models, into two internal files. These files may be translated into an explicit equation listing, a picture (matrix layout) or the MPSX input file. LPL includes also a link to the popular relational database management system dBASEIII, which allows the LPL Compiler to read data, used in the LP model, directly from databases.ZusammenfassungDieser Artikel beschreibt eine spezialisierte Programmiersprache LPL (Linear Programming Language), die sich dazu eignet, LP-Modelle aufzubauen, zu modifizieren und dokumentieren. Die LPL-Sprache wurde zum Erstellen von MPSX-Input-Dateien größerer Modelle erfolgreich eingesetzt. Der LPL-Compiler übersetzt ein LPL-Programm, das ein vollständiges LP-Model repräsentiert, in zwei interne Dateien. Diese beiden Dateien können ihrerseits wieder übersetzt werden in eine Gleichungsliste, eine Matrix-Darstellung oder in eine MPSX-Datei. LPL beinhaltet ebenfalls eine Schnittstelle zur populären Datenbank dBASEIII, welche erlaubt, Daten, die vom LP-Model gebraucht werden, direkt ab der Datenbank einzulesen.

A Modeling Framework

After having investigated and criticized various approaches in the last chapter, it is time to present my own view of a computer-based modeling framework. As one might expect, it has been inspired from many sources. The different algebraic languages have had a certain influence, although at least the initial versions of LPL were developed completely independently from any other algebraic language. Structured Modeling too, influenced me, helping to shape the point of view presented in this chapter. More conceptual and implementation specific aspects were influenced by computer science and especially programming language design, as taught by Wirth [Wirth 1996], [Wirth/Gutknecht 1992].

LPL: A mathematical programming language

SummaryThis paper describes the new version of the modeling language, named LPL (Linear Programming Language). It may be used to build, modify and document mathematical models. The LPL language has been successfully applied to generate automatically MPS input files and reports of large LP models. The available LPL compiler translates LPL programs to the input code of any LP/MIP solver, calls the solver automatically, reads the solution back to its internal representation, and the integrated Report Generator produces the user defined reports of the model. Furthermore, an Input Generator can read the data from many formats.ZusammenfassungDieser Artikel beschreibt die neue Version der Modellierungssprache LPL (Linear Programming Language), die sich dazu eignet, mathematische Modelle aufzubauen, zu warten und zu dokumentieren. Die LPL-Sprache wurde zum Erstellen von MPS-InputDateien und Resultate-Tabellen größerer LP-Modelle erfolgreich eingesetzt. Der LPL-Compiler übersetzt ein LPL-Programm, das ein vollständiges Modell repräsentiert, in den Eingabecode eines LP/MIP-Lösungsprogramms, ruft den Lösungsalgorithmus auf, liest die Lösung, und ein integrierter Tabellengenerator gibt vom Benutzer definierte Resultate-Tabellen aus. Außerdem erlaubt ein Dateneingabe-Generator, die Daten in verschiedenen Formaten zu lesen.

The LPL Modeling Language

LPL is a structured mathematical, logical modeling and programming language with a powerful index mechanism, which allows one to build, maintain, modify, and document large linear, non-linear, and other mathematical models. It is also a modeling system with a point-and-click user interface to browse, edit, and run models. LPL includes some special features: Integration of algorithmic and declarative knowledge, logical modeling, automatic generation of model documentation, and others.

An Overview of Approaches

Different approaches and modeling tools have been proposed. A quick tour is given which makes no claim of being complete. Rather the aim is to get an idea of what can be considered as the present state of mathematical modeling.

The Definition of the Language

LPL is not a commercial system which needs to compete with other systems. The language is relatively simple and small. There are three reasons why LPL has been developed. In the mid-eighties, when the LPL project began, there were virtually no modeling tools available for personal computers. Yet, people at our Institute of Informatics had to manage real-life, large models. LPL and other tools have been developed and have been used since for this purpose. The second reason was to have computer-based tools for teaching modeling in operations research. I always found it to be an anachronism to teach modeling in OR without a computer. After all, wasn’t OR a child of the advent of computers?

Problems and Concepts

From the more mathematical aspects of modeling, we now switch to the implementation aspects of computer-based modeling tools. Therefore, this chapter is more relevant for the designer of modeling tools than for the modeler or the model user. The previous chapters detail important prerequisites for understanding what modeling is, what components are necessary to implement modeling tools, and what model types should be part of the framework. In a word, from mathematics we switch to computer science. From a computer science point of view, this book could be viewed as the presentation of a large software project: The first four chapters describe the functionality and the “software requirement catalogue” from the user’s point of view. This chapter summarizes these requirements and presents the formal specifications. Finally, Part III presents a concrete implementation; the LPL system.

Index Sets in Modeling Languages

Index sets are an integral and fundamental part of every mathematical modeling language. They assist the modeler in grouping various objects and entities. Index sets are also used extensively in the mathematical notation to write an expression in a concise way. An example is the sigma notation for formulating the summation of an unknown number n of terms. In this paper, the concept of index set is introduced in the context of modeling languages. The main objective is to propose an extension and generalization of the concept of index sets, which is the concept of hierarchical index sets. The paper concludes with an application, which clearly shows the usefulness of this concept.

What is Modeling

This work is about computer-based mathematical modeling tools. But before we can implement these tools, we have to understand modeling and, above all, mathematical modeling. The purpose of this chapter is to give a precise definition of the term model. The overview will begin with general, unspecified notions, and then proceed to more formal concepts. Finally, a short historical digression will be presented to suggest further arguments for the importance of mathematical modeling.