Dries Goossens

Exact algorithms for procurement problems under a total quantity discount structure

In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the total quantity discount (TQD) problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe four variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. We also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought. In a third variant, the number of winning suppliers is limited, both in general and per product. Finally, we investigate a multi-period variant, where the buyer not only needs to decide what goods to buy from what supplier, but also when to do this, while considering the inventory costs. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.

Scheduling the Belgian Soccer League

Every sports league needs a game schedule; such a schedule is important because it influences all parties involved and even the outcome of the sports competition. Interest in Belgian soccer has increased during recent years, as has interest in sports leagues in other countries. This paper describes our experiences in scheduling the highest division in the Belgian soccer league. We describe how we automated and improved the development of the 2006--2007 season schedule, and explain how we achieved additional improvement by dividing the scheduling problem into two problems. The resulting calendar has been accepted for the 2007--2008 season. We also compare the quality of the schedules generated using different scheduling methods.

Efficient solutions for Mastermind using genetic algorithms

We present a new genetic algorithm for playing the game of Mastermind. The algorithm requires low run-times and results in a low expected number of guesses. Its performance is comparable to that of other meta-heuristics for the standard setting with four positions and six colors, while it outperforms the existing algorithms when more colors and positions are examined. The central idea underlying the algorithm is the creation of a large set of eligible guesses collected throughout the different generations of the genetic algorithm, the quality of each of which is subsequently determined based on a comparison with a selection of elements of the set.

Comparing league formats with respect to match importance in Belgian football

Recently, most clubs in the highest Belgian football division have become convinced that the format of their league should be changed. Moreover, the TV station that broadcasts the league is pleading for a more attractive competition. However, the clubs have not been able to agree on a new league format, mainly because they have conflicting interests. In this paper, we compare the current league format, and three other formats that have been considered by the Royal Belgian Football Association. We simulate the course of each of these league formats, based on historical match results. We assume that the attractiveness of a format is determined by the importance of its games; our importance measure for a game is based on the number of teams for which this game can be decisive to reach a given goal. Furthermore, we provide an overview of how each league format aligns with the expectations and interests of each type of club.

A Framework for Scheduling Professional Sports Leagues

This paper introduces a framework for a highly constrained sports scheduling problem which is modeled from the requirements of various professional sports leagues. We define a sports scheduling problem, introduce the necessary terminology and detail the constraints of the problem. A set of artificial and real‐world instances derived from the actual problems solved for the professional sports league owners are proposed. We publish the best solutions we have found, and invite the sports scheduling community to find solutions to the unsolved instances. We believe that the instances will help researchers to test the value of their solution methods. The instances are available online.

The carryover Effect Does Not Influence Football Results

In a round robin tournament, it is often believed that each team has an effect on its opponent, which carries over to the next game of that opponent. Indeed, if team A plays against team B, and subsequently against team C, A’s performance against C may have been affected by B, and we say that team C receives a carryover effect from B. For instance, if team B is a very strong team, then team A could be exhausted and discouraged after this game, which could benefit its next opponent, team C. Clearly, any schedule will lead to carryover effects. In practice, the perceived influence of carryover effects has been used as an argument when producing a schedule. In this work, we develop an approach to measure whether carryover effects have an influence on the outcome of football matches. The authors apply this method on the highest division in Belgium, using data from over 30 seasons, amounting over 10,000 matches. In our data set, we find no evidence to support the claim that carryover effects affect the results, which has major implications for the sporting community with respect to generating fixtures.

Optimizing the unlimited shift generation problem

Good rosters have many benefits for an organization, such as lower costs, more effective utilization of resources and fairer workloads. This paper introduces the unlimited shift generation problem. The problem is to construct a set of shifts such that the staff demand at each timeslot is covered by a suitable number of employees. A set of real-world instances derived from the actual problems solved for various companies is presented, along with our results. This research has contributed to better systems for our industry partners.

Using Mixed-Integer Programming to Win a Cycling Game

This paper presents an application of optimization modeling to the winning of a popular cycling game. The application includes real-life data of contempory cyclists. It also has the potential to motivate students with a competitive but fun “race” for a solution. Because the developed optimization model contains features of knapsack problems, multiperiod inventory problems, and logical constraint modeling, it is perfectly suitable for a concluding case study in an undergraduate operations research/management science course. The application was originally developed for an MBA operations research course focusing on spreadsheet modeling skills, but it can also be used in courses that focus on algebraic modeling of optimization problems.

The Red–Blue transportation problem

This paper considers the Red–Blue Transportation Problem (Red–Blue TP), a generalization of the transportation problem where supply nodes are partitioned into two sets and so-called exclusionary constraints are imposed. We encountered a special case of this problem in a hospital context, where patients need to be assigned to rooms. We establish the problem’s complexity, and we compare two integer programming formulations. Furthermore, a maximization variant of Red–Blue TP is presented, for which we propose a constant-factor approximation algorithm. We conclude with a computational study on the performance of the integer programming formulations and the approximation algorithms, by varying the problem size, the partitioning of the supply nodes, and the density of the problem.

The impact of the soccer schedule on TV viewership and stadium attendance: evidence from the Belgian Pro League.

In the past decade, television (TV) broadcasters have been investing a huge amount of money for the Belgian Pro League broadcasting rights. These companies pursue an audience rating maximization, which depends heavily on the schedule of the league matches. At the same time, clubs try to maximize their home attendance and find themselves affected by the schedule as well. Our article aims to capture the Belgian soccer fans’ preferences with respect to scheduling options, both for watching matches on TV and in the stadium. We carried out a discrete choice experiment using an online survey questionnaire distributed on a national scale. The choice sets are based on three match characteristics: month, kickoff time, and quality of the opponent. The first part of this survey concerns TV broadcasting aspects. The second part includes questions about stadium attendance. The choice data are first analyzed with a conditional logit model which assumes homogenous preferences. Then a mixed logit model is fit to model the heterogeneity among the fans. The estimates are used to calculate the expected utility of watching a Belgian Pro League match for every possible setting, either on TV or in the stadium. These predictions are validated in terms of the real audience rating and home attendance data. Our results can be used to improve the scheduling process of the Belgian Pro League in order to persuade more fans to watch the matches on TV or in a stadium.