Section: Application Domains
Academic generic problems
In this project, some well known optimization problems are revisited in terms of multiobjective modelization and resolution:

Workshop optimization problems:
Workshop optimization problems deal with optimizing the production. In this project, two specific problems are under study.

Flowshop scheduling problem: The flowshop problem is one of the most wellknown scheduling problems. However, most of the works in the literature use a monoobjective model. In general, the minimized objective is the total completion time (makespan). Many other criteria may be used to schedule tasks on different machines: maximum tardiness, total tardiness, mean job flowtime, number of delayed jobs, maximum job flowtime, etc. In the DOLPHIN project, a bicriteria model, which consists in minimizing the makespan and the total tardiness, is studied. A tricriteria flowshop problem, minimizing in addition the maximum tardiness, is also studied. It allows us to develop and test multiobjective (and not only biobjective) exact methods.

Cutting problems: Cutting problems occur when pieces of wire, steel, wood, or paper have to be cut from larger pieces. The objective is to minimize the quantity of lost material. Most of these problems derive from the classical onedimensional cuttingstock problem, which have been studied by many researchers. The problem studied by the DOLPHIN project is a twodimensional biobjective problem, where rotating a rectangular piece has an impact on the visual quality of the cutting pattern. First we have to study the structure of the cuttingstock problem when rotation is allowed, then we will develop a method dedicated to the biobjective version of the problem.


Logistics and transportation problems:

Packing problems: In logistic and transportation fields, packing problems may be a major issue in the delivery process. They arise when one wants to minimize the size of a warehouse or a cargo, the number of boxes, or the number of vehicles used to deliver a batch of items. These problems have been the subjects of many papers, but only few of them study multiobjective cases, and to our knowledge, never from an exact point of view. Such a case occurs for example when some pairs of items cannot be packed in the same bin. The DOLPHIN project is currently studying the problem in its onedimensional version. We plan to generalize our approach to two and three dimensional problems, and to more other conflict constraints, with the notion of distance between items.

Routing problems: The vehicle routing problem (VRP) is a wellknown problem and it has been studied since the end of the 50's. It has a lot of practical applications in many industrial areas (ex. transportation, logistics, etc). Existing studies of the VRP are almost all concerned with the minimization of the total distance only. The model studied in the DOLPHIN project introduces a second objective, whose purpose is to balance the length of the tours. This new criterion is expressed as the minimization of the difference between the length of the longest tour and the length of the shortest tour. As far as we know, this model is one of the pioneer work in the literature.
The second routing problem is a generalization of the covering tour problem (CTP). In the DOLPHIN project, this problem is solved as a biobjective problem where a set of constraints are modeled as an objective. The two objectives are: i) minimization of the length of the tour; ii) minimization of the largest distance between a node to be covered and a visited node. As far as we know, this study is among the first works that tackle a classic monoobjective routing problem by relaxing constraints and building a more general MOP.
The third studied routing problem is the Ring Star Problem (RSP). This problem consists in locating a simple cycle through a subset of nodes of a graph while optimizing two kinds of costs. The first objective is to minimize a ring cost that is related to the length of the cycle. The second one is to minimize an assignment cost from nonvisited nodes to visited ones. In spite of its natural bicriteria formulation, this problem has always been studied in a singleobjective form where either both objectives are combined or one objective is treated as a constraint.
Recently, within a cooperation with SOGEP, the logistic and delivery subsidiary company of REDCATS (PINAULT PRINTEMPS REDOUTE), a new routing problem is under study. Indeed, the COLIVAD project consists in solving a logistic and transportation problem that has been reduced to a vehicle routing problem with additional constraints. First we are designing a method to solve exactly a biobjective version of the problem in order to evaluate the interest of modifying the current process of delivery. We are also working on the resolution of a singleobjective version of this problem to design an operational tool dedicated to the SOGEP problem.

For all studied problems, standard benchmarks have been extended to the multiobjective case. The benchmarks and the obtained results (optimal Pareto front, best known Pareto front) are available on the Web pages associated to the project and from the MCDM (International Society on Multiple Criteria Decision Making) Web site. This is an important issue to encourage comparison experiments in the research community.