Section: New Results

Packing, Planning and Scheduling

Participants : Sophie Michel, Andrew Miller, Ruslan Sadykov, Gautier Stauffer, François Vanderbeck.

Bin-Packing and Knapsack variants

The bin-packing problem raise the qestion of the minimum number of bin of fixed size we need 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 [32] . We show 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 (we close 8 of the 10 open instances so far). The approach involves generic primal heuristics, generic branching, but a specific pricing procedure.

This specific precedure consist in a variant of a knapsack problem. This latter consist in choosing the most usefull items among a set while the total weight does not exceed a certain bound. Note that knapsack problems find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig-Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently.

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 usualy 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.

We also encountered a variant of the knapsack problem in an inventory routing problem submitted by our industrial partner Exeo Solutions [122] . This problem must construct the planning of single product pickups over time; each site accumulates stock at a deterministic rate; the stock is emptied on each visit. As a subproblem of a Dantzig-Wolfe decomposition approach applied to this inventory routing problem, we faced a multiple-class integer knapsack problem with setups [18] . Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.

Workforce Scheduling

In reply to a challenge proposed by France Telecom (the french telecommunication operator), G. Stauffer and S. Pokutta worked on a problem where the purpose is to form technician teams and to plan different tasks in order to minimize the makespan. The tasks are linked by precedence constraints and can be treated only by technician teams having the good expierience. The proposed method is described in [21] along with computationnal experiments coming from real instances.

Machine scheduling

Ruslan Sadykov in collaboration with Prof. Philippe Baptiste from the Ecole Polytechnique (Paris) has worked on the problem to schedule an airborne radar. This research has been done in the framework of a joint project between the Ecole Polytechnique and the DGA. Airborne radars are widely used to perform a large variety of tasks in a fighter aircraft. These tasks include, but are not limited to, searching, tracking and identifying targets. Such tasks play a crucial role for the aircraft and they are repeated in a “more or less” cyclic fashion. This defines a complex scheduling problem that impacts a lot on the quality of the radars output and on the overall safety of the aircraft.

For this problem, three different Mixed Integer Programming formulations have been proposed [13] . Two of the formulations are compact and can be solved directly by an MIP solver. The third formulation relies on a Branch-and-Price algorithm. Theoretical and experimental comparisons of the formulations were reported. A dedicated solver has been implemented for this problem, and real-life instances of a moderate size have been solved to optimality. This work will allow us to estimate the quality of heuristic algorithms for the problem. Only fast heuristic algorithms can be used in practice due to tight resolution time restrictions. The research in this direction has been already started  [99] .

Another study realised with Prof. Ph. Baptiste concerns the scheduling situation in which a set of jobs subjected to release dates and deadlines are to be performed on a single machine [12] . The objective is to minimize a piecewise linear objective function where F_j(C_j) corresponds to the cost of the completion of job j at time C_j . This class of function is very large and thus interesting both from a theoretical and practical point of view: It can be used to model total (weighted) completion time, total (weighted) tardiness, earliness and tardiness, etc. We introduce a new Mixed Integer Program (MIP) based on time interval decomposition. Our MIP is closely related to the well-known time-indexed MIP formulation but uses much less variables and constraints. Experiments on academic benchmarks as well as on real-life industrial problems show that our generic MIP formulation is efficient.

Another research in progress concerns scheduling parallel jobs i.e. which 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. Therefore, parallel jobs scheduling is becoming more and more widespread. In this work, we consider the NP-hard problem of scheduling a class of parallel jobs to minimize the total weighted completion time or mean weighted flow time criteria. For this problem, we have introduced an important dominance rule which can be used to reduce the search space while searching for an optimal solution [26] .

We have also studied in a scheduling problem that takes place at cross docking terminals. In such places, products from incoming trucks are sorted according to there destinations and transferred to outgoing trucks using a temporary storage. Such terminals allow companies to reduce storage and transportation costs in supply chain. This paper focuses on the operational activities at cross docking terminals. In [38] , we consider the trucks scheduling problem with the objective to minimise the storage usage during the product transfer. We show that a simplification of this NP-hard problem in which the arrival sequences of incoming and outgoing trucks are fixed is polynomially solvable by proposing a dynamic programming algorithm for it.

Multi-item production planning problems

In research recently accepted for publication ([11] ), we have developed rigorous computational methods to find high quality production plans for big bucket lot-sizing problems of realistic size. By big bucket we mean problems in which multiple product categories compete for the same capacities (of machines, labor, etc.) Our methods are simple and direct enough to be implemented in a high-level algebraic modeling language (rather than a low level programming language such as C or Fortran). Moreover, they adapt easily to different situations that may arise depending on particularities of the bill-of-materials, specific facility characteristics, etc. There is therefore hope that these methods may prove practical for simple industrial use.

In [42] , we have compared various methods for finding performance guarantees (lower bounds) for realistically sized instances of such problems. These methods include both those previously proposed in the literature and those we have developed ourselves. Our methods of comparison were both theoretical and computational; one of the primary contributions of this research is to identify and highlight those aspects of these problems that prevent us from solving them more effectively. This identification could be crucial in improving our ability to solve such models.

The following theoretical research was directly inspired by production planning problems. In [14] , the authors discuss a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. As mention before, these results directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in O(nlogn) . Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.

Energy production planning

We are currently working on a project aiming to plan the energy production and the maintenance breaks for a set of power plants generating electricity. This problem has two different levels of decisions. The first one consist in determining, for a certain time horizon, when the different power plants will have to stop in order to perform a refueling and to decide the amount of this reflueling. Given a set of scenarios defining variable levels of energy consumption, the second decision level aims to decide the quantity of power each plant will have to produce.

As the number of periods composing the time horizon, and the number of scenario are quite large, the size of any MIP formulation for such problem will forbide a exact resolution of the problem in an acceptable time. However, our objective is to show that exact methods can be used on simpler problems and combined to design heuristics for the solution of large scale problem.

Stochastic optimization of allocation problems

The allocation of surgeries to operating rooms (ORs) is a challenging combinatorial optimization problem. There is moreover significant uncertainty in the duration of surgical procedures, which further complicates assignment decisions. In [17] , we present stochastic optimization models for the assignment of surgeries to ORs on a given day of surgery. The objective includes a fixed cost of opening ORs and a variable cost of overtime relative to a fixed length-of-day. We describe two types of models. The first is a two-stage stochastic linear program with binary decisions in the first-stage and simple recourse in the second stage. The second is its robust counterpart, in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations. We describe the mathematical models, bounds on the optimal solution, and solution methodologies, including an easy-to-implement heuristic. Numerical experiments based on real data from a large health care provider are used to contrast the results for the two models, and illustrate the potential for impact in practice. Based on our numerical experimentation we find that a fast and easy-to-implement heuristic works fairly well on average across many instances. We also find that the robust method performs approximately as well as the heuristic, is much faster than solving the stochastic recourse model, and has the benefit of limiting the worst-case outcome of the recourse problem.