## Section: New Results

### Model Specific Developments and Applications

Participants : Cédric Joncour, Andrew Miller, Arnaud Pêcher, Pierre Pesneau, Ruslan Sadykov, Gautier Stauffer, Damien Trut, François Vanderbeck.

The models on which we made progress can be partitionned in three areas: “Packing and Covering Problems”, “Network Design and Routing”, and “Planning, Scheduling, and Logistic Problems”.

#### Bin-Packing and Knapsack with Conflicts

The bin-packing problem consists in finding the minimum number of bin of fixed size one needs to pack a set of items of different sizes. We studied a generalization of this problem where items can be in conflicts and thus cannot be put together in the same bin. We show in [21] that the instances of the literature with 120 to 1000 items can be solved to optimality with a generic Branch-and-Price algorithm, such as our prototype BaPCod, within competitive computing time. Moreover, we solved to optimality all the 37 open instances. The approach involves generic primal heuristics, generic branching, but a specific pricing procedure.

The knapsack variant encountered in our bin packing problem resolution considers conflicts between items. This problem is quite difficult to solve compared to the usual knapsack problem. The latter is already NP-hard, but can be usually efficiently solved by dynamic programming. We have shown that when the conflict graph (the graph defining the conflicts between the items) is an interval graph, this generalization of the knapsack can also be solved quite efficiently by dynamic programming with the same complexity than the one to solve the common knapsack problem. For the case, when the conflict graph is arbitrary, we proposed a very efficient enumeration algorithm which outperforms the approaches used in the literature.

#### Using graph theory for solving orthogonal knapsack problems

We investigated the
orthogonal knapsack problem, with the help of graph
theory. The multi-dimensional orthogonal packing problem (OPP) is
defined as follows: given a set of items with rectangular
shapes, the problem is to decide whether there is a non-overlapping
packing of these items in a rectangular bin. The rotation of items is
not allowed. A powerful caracterization of packing configurations by
means of interval graphs was introduced by Fekete and Schepers
using an efficient representation
of all geometrically symmetric solutions by a so called
*packing class* involving one *interval graph* (whose
complement admits a transitive orientation: each such orientation of
the edges corresponds to a specific placement of the forms) for each
dimension. Though Fekete & Schepers' framework is very efficient,
we have however identified several weaknesses in their algorithms:
the most obvious one is that they do not take advantage of the
different possibilities to represent interval graphs.

In [13] , [14] [57] , we give two new algorithms: the first one is based upon matrices with consecutive ones on each row as data structures and the second one uses so-called MPQ-trees, which were introduced by Korte and Möhring to recognize interval graphs. These two new algorithms are very efficient, as they outperform Fekete and Schepers' on most standard benchmarks.

#### Inventory routing and pickup-and-delivery problems

Inventory routing problems combine the optimization of product deliveries (or pickups) with inventory control at customer sites. in [15] , we considered the planning of single product pickups over time: each site accumulates stock at a deterministic rate; the stock is emptied on each visit. Our objective is to minimize a surrogate measure of routing cost while achieving some form of regional clustering by partitioning the sites between the vehicles. The fleet size is given but can potentially be reduced. Planning consists in assigning customers to vehicles in each time period, but the routing, i.e., the actual sequence in which vehicles visit customers, is considered as an “operational” decision. We developed a truncated branch-and-price algorithm. This exact optimization approach is combined with rounding and local search heuristics to yield both primal solutions and dual bounds that allow us to estimate the deviation from optimality of our solution. We were confronted with the issue of symmetry in time that naturally arises in building a cyclic schedule (cyclic permutations along the time axis define alternative solutions). Central to our approach is a state-space relaxation idea that allows us to avoid this drawback: the symmetry in time is eliminated by modelling an average behavior. Our algorithm provides solutions with reasonable deviation from optimality for large scale problems (260 customer sites, 60 time periods, 10 vehicles) coming from industry. The subproblem is interesting in its own right: it is a multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption.

Through the internship of Damien Trut, we studyied the optimization problem consisted in the planning of the pick-up of full waste container and delivery of empty container at customer sites by simple vehicles that can carry a single container, or vehicles with a trailer attached that have a total capacity of 2 containers but require more time when handling containers. The model is a multi-period, multi-vehicle, pickup and delivery problem, with “many-to-many” multi commodity transfer requirements and transhipment nodes. In its short term variant, urgent order are coming online. We developed the prototype of a branch-and-price approach for this problem. The prototype was used by Exeo to convince their customer of the potential benefit of decision aid tools to automatically generate vehicle routes. Next, we shall be considering the dimensioning of a vehicle fleet and their allocation to cluster of collect points in a periodic solution (for a PhD project).

In collaboration with the group of M-C Speranza of the university of Brescia (Italy), we study the Vehicle Routing Problem with Discrete Split Deliveries (a customer demand can be partition in integer lot assigned to different vehicles). The development is done within BaPCod with specialized pricing and branching scheme.

#### Time-Dependent Travelling Salesman Problem and Resource Constrained Shortest Path

In [12] we present a new formulation for the Time-Dependent Travelling Salesman Problem (TDTSP). The main feature of our formulation is that it uses, as a subproblem, an exact description of the n-circuit problem. We present a new extended formulation that is based on using, for each node, a stronger subproblem, namely a $n$-circuit subproblem with the additional constraint that the corresponding node is not repeated in the circuit. Although the new model has more variables and constraints than the model of Picard and Queyranne (1978), the results given from our computational experiments show that the linear programming relaxation of the new model gives, for many of the instances tested, gaps that are close to zero. We also provided a complete characterization of the feasible set of the corresponding linear programming relaxation in the space of the variables of the PQ model.

Following this work, we proposed an extended formulation in terms of the Asymmetric Travelling Salesman Problem (ATSP) in [33] . A tightening the linear programming relaxation is obtained by i) enhancing the subproblem arising in the standard multicommodity flow (MCF) model for the ATSP and then ii) by using modelling enhancement techniques. We compare the linear programming relaxation of the new formulation with the linear programming relaxation of the three compact and non-dominated formulations presented in Oncan et al. (2009). As a result of this comparison we present an updated classification of formulations for the asymmetric traveling salesman problem (ATSP).

In the intership of André Linhares, we studied the Resource Constrained Shortest Path Problem (RCSPP): we presented some of the state-of-the-art dynamic programming methods for solving the RCSPP in a unified manner, and we proposed some variants of these algorithms. We assessed the effectiveness of these algorithms through computational experiments.

#### Machine scheduling

The column-and-row generation method presented in [28] , [25] is quite effective for the general machine scheduling problem. In our work [29] , we show indeed that one of the most efficient approaches to solve this problem is to use time-indexed Integer Programming formulation, and to deal with its huge size by generating variables and constraints dynamically. The numerical results of [29] , highlight the significant reduction in computing times that results from applying the column-and-row generation approach.

In [20] , we coonsider the scheduling jobs in parallel, i.e., jobs can be executed on more than one processor at the same time. With the emergence of new production, communication and parallel computing system, the usual scheduling requirement that a job is executed only on one processor has become, in many cases, obsolete and unfounded. In this work, we consider the NP-hard problem of scheduling malleable jobs to minimize the total weighted completion time (or mean weighted flow time). For this problem, we introduce the class of “ascending” schedules in which, for each job, the number of machines assigned to it cannot decrease over time while this job is being processed. We prove that, under a natural assumption on the processing time functions of jobs, the set of ascending schedules is dominant for the problem. This result can be used to reduce the search space while looking for an optimal solution.

#### One warehouse multi-retailer problem

The *One-Warehouse Multi-retailer problem (OWMR)* is a very
important NP-hard inventory control problem arising in the
distribution of goods when one central warehouse is supplying a set of
final retailers facing demand from customers. In
[30] , we provide a simple and fast
2-approximation algorithm for this problem (i.e. an algorithm ensuring
a deviation by a factor at most two from the optimal solution). This
result is both important in practice and in theory as it allows to
approximate large real-world instances of the problem (we implemented
this algorithm at IBM and it is within 10% of optimality in practice)
and the techniques we developed appear to apply to more general
settings. We are extending our results to other inventory control
problems.