In retail, assortment planning refers to selecting a subset of products to offer that maximizes profit. Assortments can be planned for a single store or a retailer with multiple chain stores where demand varies between stores. In this paper, we assume that a retailer with a multitude of stores wants to specify her offered assortment. To suit all local preferences, regionalization and store-level assortment optimization are widely used in practice and lead to competitive advantages. When selecting regionalized assortments, a tradeoff between expensive, customized assortments in every store and inexpensive, identical assortments in all stores that neglect demand variation is preferable.
We formulate a stylized model for the regionalized assortment planning problem (APP) with capacity constraints and given demand. In our approach, a 'common assortment' that is supplemented by regionalized products is selected. While products in the common assortment are offered in all stores, products in the local assortments are customized and vary from store to store.
Concerning the computational complexity, we show that the APP is strongly NP-complete. The core of this hardness result lies in the selection of the common assortment. We formulate the APP as an integer program and provide algorithms and methods for obtaining approximate solutions and solving large-scale instances.
Lastly, we perform computational experiments to analyze the benefits of regionalized assortment planning depending on the variation in customer demands between stores.
We consider the problem to evacuate several regions due to river flooding, where sufficient time is given to plan ahead. To ensure a smooth evacuation procedure, our model includes the decision which regions to assign to which shelter, and when evacuation orders should be issued, such that roads do not become congested.
Due to uncertainty in weather forecast, several possible scenarios are simultaneously considered in a robust optimization framework. To solve the resulting integer program, we apply a Tabu search algorithm based on decomposing the problem into better tractable subproblems. Computational experiments on random instances and an instance based on Kulmbach, Germany, data show considerable improvement compared to an MIP solver provided with a strong starting solution.
We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker.
We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding.
In this paper we consider the problem of decomposing a given integer matrix A into
a positive integer linear combination of consecutive-ones matrices with a bound on the
number of columns per matrix. This problem is of relevance in the realization stage
of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf
collimators with limited width. Constrained and unconstrained versions of the problem
with the objectives of minimizing beam-on time and decomposition cardinality are considered.
We introduce a new approach which can be used to find the minimum beam-on
time for both constrained and unconstrained versions of the problem. The decomposition
cardinality problem is shown to be NP-hard and an approach is proposed to solve the
lexicographic decomposition problem of minimizing the decomposition cardinality subject
to optimal beam-on time.
We consider storage loading problems where items with uncertain weights have
to be loaded into a storage area, taking into account stacking and
payload constraints. Following the robust optimization paradigm, we propose
strict and adjustable optimization models for finite and interval-based
uncertainties. To solve these problems, exact decomposition and heuristic
solution algorithms are developed.
For strict robustness, we also present a compact formulation based
on a characterization of worst-case scenarios.
Computational results show that computation times and algorithm
gaps are reasonable for practical applications.
Furthermore, we find that the robustness concepts show different
potential depending on the type of data being used.
Scheduling-Location (ScheLoc) Problems integrate the separate fields of
scheduling and location problems. In ScheLoc Problems the objective is to
find locations for the machines and a schedule for each machine subject to
some production and location constraints such that some scheduling object-
ive is minimized. In this paper we consider the Discrete Parallel Machine
Makespan (DPMM) ScheLoc Problem where the set of possible machine loc-
ations is discrete and a set of n jobs has to be taken to the machines and
processed such that the makespan is minimized. Since the separate location
and scheduling problem are both NP-hard, so is the corresponding ScheLoc
Problem. Therefore, we propose an integer programming formulation and
different versions of clustering heuristics, where jobs are split into clusters
and each cluster is assigned to one of the possible machine locations. Since
the IP formulation can only be solved for small scale instances we propose
several lower bounds to measure the quality of the clustering heuristics. Ex-
tensive computational tests show the efficiency of the heuristics.
Motivated by the time-dependent location problem over T time-periods introduced in
Maier and Hamacher (2015) we consider the special case of two time-steps, which was shown
to be equivalent to the static 2-facility location problem in the plane. Geometric optimality
conditions are stated for the median objective. When using block norms, these conditions
are used to derive a polygon grid inducing a subdivision of the plane based on normal cones,
yielding a new approach to solve the 2-facility location problem in polynomial time. Combinatorial algorithms for the 2-facility location problem based on geometric properties are
deduced and their complexities are analyzed. These methods differ from others as they are
completely working on geometric objects to derive the optimal solution set.
A single facility problem in the plane is considered, where an optimal location has to be
identified for each of finitely many time-steps with respect to time-dependent weights and
demand points. It is shown that the median objective can be reduced to a special case of the
static multifacility median problem such that results from the latter can be used to tackle the
dynamic location problem. When using block norms as distance measure between facilities,
a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the
resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation
between dynamic location problems for T time periods and T-facility problems, this algorithm
can also be applied to the static 2-facility location problem.
Due to the increasing number of natural or man-made disasters, the application of operations research methods in evacuation planning has seen a rising interest in the research community. From the beginning, evacuation planning has been highly focused on car-based evacuation. Recently, also the evacuation of transit depended evacuees with the help of buses has been considered.
In this case study, we apply two such models and solution algorithms to evacuate a core part of the metropolitan capital city Kathmandu of Nepal as a hypothetical endangered region, where a large part of population is transit dependent. We discuss the computational results for evacuation time under a broad range of possible scenarios, and derive planning suggestions for practitioners.
A new algorithm for optimization problems with three objective functions is presented which computes a representation for the set of nondominated points. This representation is guaranteed to have a desired coverage error and a bound on the number of iterations needed by the algorithm to meet this coverage error is derived. Since the representation does not necessarily contain nondominated points only, ideas to calculate bounds for the representation error are given. Moreover, the incorporation of domination during the algorithm and other quality measures are discussed.