Evaluation of a new supply strategy based on stochastic programming for a fashion discounter

Fashion discounters face the problem of ordering the right amount of pieces in each size of a product. The product is ordered in pre-packs containing a certain size-mix of a product. For this so-called lot-type design problem, a stochastic mixed integer linear programm was developed, in which price cuts serve as recourse action for oversupply. Our goal is to answer the question, whether the resulting supply strategy leads to a supply that is significantly more consistent with the demand for sizes compared to the original manual planning. Since the total profit is influenced by too many factors unrelated to sizes (like the popularity of the product, the weather or a changing economic situation), we suggest a comparison method which excludes many outer effects by construction. We apply the method to a real-world field study: The improvements in the size distributions of the supply are significant.Comment: 5 pages, 1 tabl

Estudo prático de regularidade de problemas de programação semidefinida

Mestrado em Matemática e Aplicações - Matemática Empresarial e TecnológicaUm problema linear de Programa c~ao Semide nida (SDP) consiste na minimiza c~ao de uma fun c~ao linear sujeita a condi c~ao de que a fun c~ao matricial linear seja semide nida. Um problema de SDP considera-se regular se certas condi c~oes est~ao satisfeitas. H a diferentes caracteriza c~oes de regularidade de um problema, sendo uma delas a veri ca c~ao da condi c~ao de Slater. Os problemas regulares de SDP t^em sido estudados e as condi c~oes de optimalidade para estes problemas t^em a forma de teoremas cl assicos do tipo Karush-Kuhn-Tucker, e s~ao facilmente veri cadas. Na pr atica, e frequente encontrar problemas n~ao regulares. O estudo destes problemas e bem mais complicado. Por isso, tem surgido o interesse em estudar e testar a regularidade dos problemas de SDP e deduzir condi c~oes de optimalidade e m etodos de resolu c~ao dos problemas n~ao regulares. Em Kostyukova e Tchemisova [32] e proposto um algoritmo, chamado Algoritmo DIIS (Algorithm of Determination of the Immobile Index Subspace), que permite veri car se as restri c~oes de um dado problema de SDP satisfazem a condi c~ao de Slater. A teoria que serve de base a constru c~ao deste algoritmo assenta nas no c~oes de ndices e subespa co de ndices im oveis, originalmente usadas em Programa c~ao Semi-In nita (SIP), e transpostas em [32] para SDP. Este algoritmo constr oi uma matriz b asica do subespa co de ndices im oveis, caso a condi c~ao de Slater n~ao seja veri cada. A dimens~ao desta matriz caracteriza o grau de n~ao regularidade do problema. O objectivo deste trabalho e estudar o Algoritmo DIIS, implement a- -lo e test a-lo usando v arios problemas de teste de diferentes bases de dados de problemas de SDP. O Algoritmo DIIS foi implementado e executado a partir do MatLab e os testes num ericos efectuados permitiram concluir que o programa constru do veri ca com sucesso a maioria dos problemas teste. Al em disso, o algoritmo permite caracterizar o grau de n~ao regularidade dos problemas de SDP e pode ser usado para constru c~ao de algoritmos de resolu c~ao dos problemas de SDP n~ao regulares.A linear problem of Semide nite Programming (SDP) consists of minimizing a linear function subject to the constraint that the linear matrix function is semide nite. An SDP problem is considered regular if certain conditions are satis- ed. There are several characterizations of a problem regularity, one of which is checking the Slater condition. The regular SDP problems have been studied and the optimality conditions for these problems have the form of Karush-Kuhn-Tucker type theorems, and are easily veri ed. In practice it is common to nd problems that are not regular. The study of these problems is far more complicated. Therefore, there has been interest in studying and testing the regularity of the problems and deduct the SDP optimality conditions and methods for solving non-regular problems. In Kostyukova e Tchemisova [32] is proposed an algorithm, called Algorithm DIIS (Algorithm of Determination of the Immobile Index Subspace), which allows to check if the constraints of a given SDP problem satisfy the Slater condition. The theory that underlies the construction of this algorithm is based on the notions of subspace of immobile indices and immobile indices properties, originally used in Semi-In nite Programming (SIP), and implemented in [32] for SDP. This algorithm constructs a basic matrix of the subspace of immobile indices if the Slater condition is not veri ed. The size of this matrix characterizes the degree of non-regularity of the problem. The purpose of this work is to study the DIIS algorithm, implement and test it using several test problems of di erent databases of SDP problems. The DIIS algorithm was implemented and executed from MatLab and numerical tests carried out showed that the program checks successfully the majority of test problems. Moreover, the algorithm allows to characterize the degree of non-regular problems of SDP and can be used to construct algorithms for solving non-regular SDP problems