Mathematical programming approaches to pricing problems

2012/2013There are many real cases where a company needs to determine the price of its products so as to maximise its revenue or profit. To do so, the company must consider customers’ reactions to these prices, as they may refuse to buy a given product or service if its price is too high. This is commonly known in literature as a pricing problem. This class of problems, which is typically bilevel, was first studied in the 1990s and is NP-hard, although polynomial algorithms do exist for some particular cases. Many questions are still open on this subject. The aim of this thesis is to investigate mathematical properties of pricing problems, in order to find structural properties, formulations and solution methods that are as efficient as possible. In particular, we focus our attention on pricing problems over a network. In this framework, an authority owns a subset of arcs and imposes tolls on them, in an attempt to maximise his/her revenue, while users travel on the network, seeking for their minimum cost path. First, we provide a detailed review of the state of the art on bilevel pricing problems. Then, we consider a particular case where the authority is using an unit toll scheme on his/her subset of arcs, imposing either the same toll on all of them, or a toll proportional to a given parameter particular to each arc (for instance a per kilometre toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial. We then address a robust approach taking into account uncertainty on parameters. We solve some polynomial cases of the pricing problem where uncertainty is considered using an interval representation. Finally, we focus on another particular case where toll arcs are connected such that they constitute a path, as occurs on highways. We develop a Dantzig-Wolfe reformulation and present a Branch-and-Cut-and-Price algorithm to solve it. Several improvements are proposed, both for the column generation algorithm used to solve the linear relaxation and for the branching part used to find integer solutions. Numerical results are also presented to highlight the efficiency of the proposed strategies. This problem is proved to be APX-hard and a theoretical comparison between our model and another one from the literature is carried out.Un problème classique pour une compagnie est la tarification de ses produits à vendre sur le marché, de façon à maximiser les revenus. Dans ce contexte, il est important que la société prenne en compte le comportement de ses clients potentiels, puisque si le prix est trop élevé, ils peuvent décider de ne rien acheter. Ce problème est communément connu dans la littérature comme un problème de tarification ou "pricing". Une approche de programmation biniveau pour ce problème a été introduite dans les années 90, révélant sa difficulté. Cependant, certains cas particuliers peuvent être résolus par des algorithmes polynomiaux, et il y a encore de nombreuses questions ouvertes sur le sujet. Cette thèse de doctorat porte sur les propriétés mathématiques des problèmes de tarification, fixant l’objectif de déterminer différentes formulations et méthodes de résolution les plus efficaces possibles, en se concentrant sur les problèmes appliqués aux réseaux de différents types. Dans les problèmes de tarification sur réseau, nous avons deux entités : une autorité qui possède un certain sous-ensemble d’arcs, et impose des péages, avec l’intention de maximiser les revenus provenant de celle-ci, et des utilisateurs qui choisissent leur chemin de moindre coût sur l’ensemble du réseau. Dans la première partie de la thèse une analyse détaillée de l’état de l’art sur les problèmes de tarification biniveau est présentée, suivie, dans la deuxième partie, par une analyse de cas particuliers polynomiaux. En particulier, nous considérons le cas où l’autorité utilise un péage unitaire sur son sous-ensemble d’arcs, soit en choisissant le même péage sur chaque arc, soit en choisissant un péage proportionnel à un paramètre donné pour chaque arc (par exemple, un péage par kilomètre). Dans le premier cas de péages égaux, il est démontré que la complexité du problème est polynomiale, tandis que dans le second cas de péages proportionnels, elle est pseudo-polynomiale. Ensuite, nous présentons une première approche d’optimisation robuste pour les problèmes de tarification sur réseau, de manière à inclure de l’incertitude sur la valeur exacte des paramètres dans le modèle, qui est typique dans les problèmes réels. Cette incertitude est représentée en utilisant des intervalles pour les paramètres et nous proposons, pour certains cas, des algorithmes de résolution polynomiaux. La troisième et dernière partie de la thèse concerne un cas difficile, le problème de tarification sur réseau dans lequel les arcs sont connectés de manière à constituer un chemin, comme c’est le cas pour les autoroutes. Initialement, nous prouvons que ce problème est APX-dur, renforçant le résultat connu jusqu’à maintenant. Ensuite, nous présentons des nouvelles formulations plus fortes, et en particulier, nous développons une reformulation de type Danztig-Wolfe, résolue par un algorithme de Branch-and-Cut-and-Price. Enfin, nous proposons différentes stratégies pour améliorer les performances de l’algorithme, pour ce qui concerne l’algorithme de génération de colonnes utilisé pour résoudre la relaxation linéaire, et pour ce qui concerne la résolution du problème avec variables binaires. Les résultats numériques complètent les résultats théoriques, en mettant en évidence l’efficacité des stratégies proposées.Un classico problema aziendale è la determinazione del prezzo dei prodotti da vendere sul mercato, in modo tale da massimizzare le entrate che ne deriveranno. In tale contesto è importante che l’azienda tenga in considerazione il comportamento dei propri potenziali clienti, in quanto questi ultimi potrebbero ritenere che il prezzo sia troppo alto e decidere dunque di non acquistare. Questo problema è comunemente noto in letteratura come problema di tariffazione o di “pricing”. Tale problema è stato studiato negli anni novanta mediante un approccio bilivello, rivelandone l’alta complessità computazionale. Tuttavia alcuni casi particolari possono essere risolti mediante algoritmi polinomiali, e ci sono sono ancora molte domande aperte sull’argomento. Questa tesi di dottorato si focalizza sulle proprietà matematiche dei problemi di tariffazione, ponendosi l’obiettivo di determinarne formulazioni e metodi risolutivi più efficienti possibili, concentrandosi sui problemi applicati a reti di vario tipo. Nei problemi di tariffazione su rete si hanno due entità: un’autorità che possiede un certo sottoinsieme di archi e vi impone dei pedaggi, con l’intento di massimizzare le entrate che ne derivano, e gli utenti che scelgono il proprio percorso a costo minimo sulla rete complessiva (a pedaggio e non). Nella prima parte della tesi viene affrontata una dettagliata analisi dello stato dell’arte sui problemi di tariffazione bilivello, seguita, nella seconda parte, dall’analisi di particolari casi polinomiali del problema. In particolare si considera il caso in cui l’autorità utilizza uno schema di pedaggio unitario sul suo sottoinsieme di archi, imponendo o lo stesso pedaggio su ogni arco, o un pedaggio proporzionale a un dato parametro relativo ad ogni arco (ad esempio un pedaggio al chilometro). Nel primo caso di pedaggi uguali, si dimostra che la complessità del problema è polinomiale, mentre nel secondo caso di pedaggi proporzionali è pseudo-polinomiale. In seguito viene affrontato un approccio di ottimizzazione robusta per alcuni problemi di tariffazione su rete, in modo da includere nei modelli un’incertezza sul valore esatto dei parametri,tipica dei problemi reali. Tale incertezza viene rappresentata vincolando i parametri in degli intervalli e si propongono, per alcuni casi, algoritmi risolutivi polinomiali. La terza e ultima parte della tesi riguarda un caso computazionalmente difficile, in cui gli archi tariffabili sono connessi in modo tale da costituire un cammino, come avviene per le autostrade. Inizialmente si dimostra che tale problema è APX-hard, rafforzando il risultato finora conosciuto. In seguito si considerano formulazioni piùforti, e in particolare si sviluppa una riformulazione di Danztig-Wolfe, risolta tramite un algoritmo di Branch-and-Cut-and-Price. Infine si propongono diverse strategie per migliorare le performance dell’algoritmo, sia per quanto riguarda l’algoritmo di generazione di colonne utilizzato per risolvere il rilassamento lineare, sia per quanto riguarda la risoluzione del problema con variabili binarie. Risultati numerici complementano quelli teorici ed evidenziano l’efficacia delle strategie proposte.XXV Ciclo198

Pricing toll roads under uncertainty

We study the toll pricing problem when the non-toll costs on the network are not fixed and can vary over time. We assume that users who take their decisions, after the tolls are fixed, have full information of all costs before making their decision. Toll-setter, on the other hand, do not have any information of the future costs on the network. The only information toll-setter have is historical information (sample) of the network costs. In this work we study this problem on parallel networks and networks with few number of paths in single origin-destination setting. We formulate toll-setting problem in this setting as a distributionally robust optimization problem and propose a method to solve to it. We illustrate the usefulness of our approach by doing numerical experiments using a parallel network

A branch-and-price algorithm for the aperiodic multi-period service scheduling problem

This paper considers the multi-period service scheduling problem with an aperiodic service policy. In this problem, a set of customers who periodically require service over a finite time horizon is given. To satisfy the service demands, a set of operators is given, each with a fixed capacity in terms of the number of customers that can be served per period. With an aperiodic policy, customers may be served before the period were the service would be due. Two criteria are jointly considered in this problem: the total number of operators, and the total number of ahead-of-time periods. The task is to determine the service periods for each customer in such a way that the service requests of the customers are fulfilled and both criteria are minimized. A new integer programming formulation is proposed, which outperforms an existing formulation. Since the computational effort required to obtain solutions considerably increases with the size of the instances, we also present a reformulation suitable for column generation, which is then integrated within a branch-and-price algorithm. Computational experiments highlight the efficiency of this algorithm for the larger instances.Peer ReviewedPostprint (author's final draft

Peak-load pricing for the European Air Traffic Management system using modulation of en-route charges

This paper extends the use of peak-load pricing (PLP) to the context of the European Air Traffic Management system, as EU regulation No 391/2013 allows the modulation of en-route charges to avoid network capacity-demand imbalance in a specific area or on a specific route at specific times. In particular, we propose a centralised approach to PLP (CPLP) where a Central Planner (CP) is responsible for setting en-route charges on the network and Airspace Users (AUs) assess the routing of each flight. Set en-route charges should guarantee that air navigation service providers (ANSPs) are able to recover their operational costs, and that AUs perform their flights avoiding imbalances between demand and available airspace capacity. Like in the current charging system, in CPLP AUs react to en-route charges (which are imposed by CP instead of ANSPs) by choosing alternative and cheaper routes. Hence, we model this relationship between the CP and the AUs as a Stackelberg game where a leader (CP) makes his/her decision first, with complete knowledge on how the follower(s) (AUs) would react to it. The Stackelberg equilibrium is obtained by means of an optimisation problem formulated as a bilevel mixed-integer linear programming model, where the CP sets, for each ANSP, one peak and one off-peak en-route charge and the AUs make their routing choice. Preliminary results on real data instances on a regional scale are presented

A branch-and-price approach for Pure Parsimony haplotyping

This thesis comes as the result of a detailed study of decomposition methods for large-scale problems and their application to a particular problem arising in computational biology. The improvements on computer capabilities and programming techniques in the last decades have widened the set of problems that can be easily solved as Mixed Integer Linear programs. However, several applications still require formulations that involve a non-tractable amount of data necessary to describe the geometry of the solution space. In these cases, decomposition methods are used to reduce the size of the problems to be addressed. In this thesis we propose the application of some of these methods, as Dantzig-Wolfe reformulation, column generation and Lagrangian relaxation, to a problem related to the study of the human genome. The human DNA is made of two double chains, each of which consists in a sequence of nucleotides. Among these, the ones related to the Single Nucleotide Polymorphisms (SNPs) are interesting as they describe the differences between individuals. We define a haplotype as a sequence of nucleotides that describes a portion of the SNPs found in a particular chromosome, and a genotype as the sequence that aggregates the information on SNPs coming from the double DNA chain of an individual. The problem we address falls into the class defining the Haplotyping Inference problem, that consists in recovering the structure of the haplotypes, given the information on the genotypes. In particular, we consider the parsimony criterion, which means that we want to find the minimum number of haplotypes able to explain all the genotypes. This problem is known to be APX-hard. There are several contributions in the literature that can be divided into two main different classes of mixed integer linear formulations. The first one presents a polynomial number of both variables and constraints, thus these formulations are solved using a branch-and-cut approach. The second class consists of formulations that present an exponential number of constraints and variables, solved with a branch-and-cut-and-price approach. The scope of this thesis is to investigate how a new formulation that involves an exponential number of variables and a polynomial number of constraints can be solved by a branch-and-price approach. Its aim is to provide a competitive algorithm with respect to other formulations from the literature, in particular those with a polynomial number of constraints and variables. We start by providing a review of the state of the art on the Haplotype Inference problem, with particular focus on the Mixed Integer Linear programming approaches for the Haplotype Inference by Pure Parsimony (HIPP) problem. We then consider a new mathematical programming formulation for HIPP that includes a set of quadratic constraints. By applying Dantzig-Wolfe reformulation, we obtained a new integer linear programming formulation, presenting an exponential number of variables and a polynomial number of constraints on the input data. This model is the basis for the development of a branch-and-price approach. Due to the large number of variables involved, a column-generation approach is needed to solve the linear relaxation at a generic node of the search tree. An initial feasible solution is easily found by means of heuristics and used as starting point to build the Restricted Master Problem (RMP). In order to find variables to be added to the RMP, we solve a dedicated subproblem, the pricing problem, that in our case presents a quadratic objective function. We propose different ways of solving the pricing problem. Among the exact methods, we consider the integer linear model obtained by linearizing the quadratic objective function and a Smart Enumeration approach, that partitions the set of feasible solutions and solves the pricing problem restricted to each subset, exploiting some extra available information to further reduce the size of the subproblems. As heuristic approaches, we at first note that the pricing problem is easily solved for particular haplotypes. Then, for investigating the remaining solutions we propose a local search-based heuristic and an Early-terminated Smart Enumeration, where we stop the Smart Enumeration approach as soon as we find a variable that can be added to the RMP. The oscillatory behaviour of the dual variables involved in the definition of the pricing problem is limited by introducing a stabilization technique adapted to our formulation. In particular, we extended the proof of convergence of this procedure, that consists in using dual values obtained as convex combinations between real dual variables and a chosen stability center, to the cases in which the stabilized dual variables are feasible for the dual problem. In order to solve the integer model, the solution of the linear relaxation is embedded in a branch-and-price approach. The branching rule we present is inspired to the well-known Ryan-Foster branching rule for set-partitioning problems. The correctness of our approach has been proved. Further observations on the similarity of the formulation's constraints to multiple set-covering ones suggest that we can relax a family of constraints to obtain a new formulation similar to a multiple set-covering. However, we note that the proposed branch-and-price algorithm applied to this formulation does not provide a feasible solution for HIPP, thus we need to integrate the proposed branching rule and recover a feasible optimal solution for HIPP. This branch-and-price approach has been implemented in C++, with the aid of SCIP libraries and Cplex solver. Results have been obtained from different classes of instances found in literature, coming from real biological data and generated using ad-hoc programs, as well as newly generated ones. The branch-and-price approach proposed for our formulation proves to be competitive with state-of-the-art polynomial-sized formulations. In fact, we can note how the linear relaxation of our formulation is tighter than other linear relaxations and provides an effective starting solution for the branch-and-price algorithm. Results show how our approach is efficient, in particular on the set of instances that contain a larger number of genotypes We proved therefore that a branch-and-price procedure provides a good solution approach for a formulation with exponential number of variables and polynomial number of constraints. Further work may include enhancements on the implementation details, such as exploring different ways of ordering the genotypes or combining heuristic and exact methods in the stabilized framework to solve the pricing problem. Moreover, it is possible to investigate the generalization of the proposed approach in order to solve set-partitioning problems

Models and algorithms for an efficient market-driven management of European airspace demand

La congestione dello spazio aereo in Europa \ue8 un fenomeno in rapida crescita che ogni anno causa ritardi e costi per tutti gli attori coinvolti. Il presente lavoro di ricerca analizza la possibilit\ue0 di contenere questo fenomeno mediante meccanismi di tipo market-driven, nello specifico legati alla modulazione delle tasse di sorvolo, che riducano la congestione redistribuendo in maniera pi\uf9 sostenibile il traffico. A tal fine un modello di ottimizzazione bilivello \ue8 stato sviluppato, che rappresenta l'interazione tra Network Manager, Air Navigation Service Provider e compagnie aeree. Tramite tale modello \ue8 possibile simulare la reazione delle compagnie aeree a diverse strategie di peak load pricing. Il modello \ue8 stato testato su istanze di dati ottenute da traffico reale, per ottenere le quali \ue8 stata sviluppata una base di dati ad hoc. I risultati ottenuti confermano la validit\ue0 della metodologia proposta e aprono numerosi spunti per possibili sviluppi futuri

An Evolutionary Multi-Objective Optimization Framework for Bi-level Problems

Genetic algorithms (GA) are stochastic optimization methods inspired by the evolutionist theory on the origin of species and natural selection. They are able to achieve good exploration of the solution space and accurate convergence toward the global optimal solution. GAs are highly modular and easily adaptable to specific real-world problems which makes them one of the most efficient available numerical optimization methods. This work presents an optimization framework based on the Multi-Objective Genetic Algorithm for Structured Inputs (MOGASI) which combines modules and operators with specialized routines aimed at achieving enhanced performance on specific types of problems. MOGASI has dedicated methods for handling various types of data structures present in an optimization problem as well as a pre-processing phase aimed at restricting the problem domain and reducing problem complexity. It has been extensively tested against a set of benchmarks well-known in literature and compared to a selection of state-of-the-art GAs. Furthermore, the algorithm framework was extended and adapted to be applied to Bi-level Programming Problems (BPP). These are hierarchical optimization problems where the optimal solution of the bottom-level constitutes part of the top-level constraints. One of the most promising methods for handling BPPs with metaheuristics is the so-called "nested" approach. A framework extension is performed to support this kind of approach. This strategy and its effectiveness are shown on two real-world BPPs, both falling in the category of pricing problems. The first application is the Network Pricing Problem (NPP) that concerns the setting of road network tolls by an authority that tries to maximize its profit whereas users traveling on the network try to minimize their costs. A set of instances is generated to compare the optimization results of an exact solver with the MOGASI bi-level nested approach and identify the problem sizes where the latter performs best. The second application is the Peak-load Pricing (PLP) Problem. The PLP problem is aimed at investigating the possibilities for mitigating European air traffic congestion. The PLP problem is reformulated as a multi-objective BPP and solved with the MOGASI nested approach. The target is to modulate charges imposed on airspace users so as to redistribute air traffic at the European level. A large scale instance based on real air traffic data on the entire European airspace is solved. Results show that significant improvements in traffic distribution in terms of both schedule displacement and air space sector load can be achieved through this simple, en-route charge modulation scheme