Batch Deterministic and Stochastic Petri Nets and Transformation Analysis Methods

International audienc

Hybrid Optimisation Method for the Facility Layout Problem

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Linearization and Decomposition Methods for Large Scale Stochastic Inventory Routing Problem with Service Level Constraints

A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, for a depot to determine delivery volumes to its customers in each period, and vehicle routes to distribute the delivery volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicles if necessary. The objective of the problem is to minimize the total inventory and transportation cost while some constraints are given to satisfy other criteria, such as the service level to limit the stockout probability at each customer and the service level to limit the overfilling probability of the warehouse of each customer. In order to tackle the SIRPSD with notorious computational complexity, we propose for it an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model can be found by the approach, and then be used to construct a near optimal solution of the SIRPSD. Numerical examples show that, for an instance of the problem with 200 customers and 5 periods that contains about 400 thousands decision variables where half of them are integer, our approach can obtain high quality near optimal solutions with a reasonable computational time on an ordinary PC

Sequencing of parts in a robotic cell

This paper considers scheduling problems in a robotic cell which produces a set of parts on several machines served by a robot. We study the problem of sequencing parts in the cell in order to minimize the production cycle time when the sequence of the robot moves is given. This problem is NP-hard for most of the one-unit robot move cycles in a robotic cell with more than two machines producing more than two part-types. We first give a mathematical formulation to the problem, and then propose a branch-and-bound algorithm to solve it. The bounding scheme of this algorithm is based on relaxing, for all the machines except two, the constraint that the machine is occupied by a part for a period at least as long as the processing time of the part. It turns out that the lower bound obtained in this way is tight. This relaxation allows us to overcome the complexity of the problem. The lower bound can be computed using the algorithm of Gilmory and Gilmore. Computational experiments on part sequencing problems in three-machine robotic cells are given

Job Shop Using Lagrangian Relaxation

Lagrangian relaxation has recently emerged as an important method for solving complex scheduling problems. The technique has succeesfully been used to obtain near-optimal solutions for one machine scheduling problems and parallel machine scheduling problems. The approach consists of relaxing the capacity constraints on machines by using Lagrangian multipliers. The relaxed problem can be decomposed into independent job level subproblems. Peter B. Luh and his colleagues extended the technique to general job shop scheduling problems by introducing more Lagrangian multipliers to relax the precedence constraints among operations. Such that each job level relaxed subproblem can be further decomposed into a set of operation level subproblems which can easily be solved by enumeration. Unfortunately, the operation level subproblems exhibit solution oscillation from iteration to iteration and, in many cases, prevent convergence of the algorithm. Although they have proposed several method to prevent solution from oscillation, none of the methods is really satisfactory. In this paper, we propose an efficient pseudopolynomial time dynamic programming algorithm to relaxed job level subproblems. We show that by extending the technique to job shop scheduling problems, the relaxation of the precedence constraints is un-necessary, and thus the oscillation problem vanishes. This algorithm also results in a much more efficient Lagrangian relaxation approach to job-shop scheduling problems. Furthermore, this algorithm makes it possible to optimize "min-max" criteria by Lagrangian relaxation. These criteria have been neglected in Lagrangian relaxation litterature for sake of indecomposability. Computational results on randomly generated problems are given to demonstrate the efficiency of the algorithm

A branch and bound approach for earliness-tardiness scheduling problems with different due dates

The problem of scheduling n jobs on a single machine in order to minimize the weighted sum of earliness and tardiness in NP-complete when jobs have different due dates. In most of the papers dedicated to this problem, authors assume that there is no idle time between two consecutive jobs. However, as indicated by several authors, this assumption is not consistent with the earliness-tardiness criterion. It is the reason why we do not make this assumption in this paper. To reach an optimal solution, we propose a branch-and-bound approach which takes advantage of some dominance properties and lower bounding procedures. Numerical experiments show that the algorithm can solve this problem with up to twenty jobs in a reasonable amont of time

Cyclic Scheduling of a Hoist with time window constraints

This paper proposes a model and a related algorithm for generating optimal cyclic schedules of hoist moves with time window constraints in a printed circuit board (PCB) electroplating facility. The algorithm is based on the branch and bound approach and requires the solution of a specific class of linear programming problems (LPPs). These LPPs are equivalent to the problems of the cycle time evaluation in bi-valued graphs. Computational experience is presented to compare the results obtained using this new algorithm with the ones proposed in the literature

Large scale stochastic inventory routing problems with split delivery and service level constraints

A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC

Cross-Evaluation Cost Allocation for Vehicle Routing Games

International audienceIn this paper, we study a cost allocation problem that arises in goods distribution by vehicles. This problem can be formulated as a vehicle-routing game (VRG), where the total distribution cost must be divided among all customers visited by the vehicles. We apply our recently developed solution concept for cooperative games, cross-evaluation value, to the cost allocation. In the solution concept, the allocation of the total cost of a VRG among its players is based on self-evaluation and peer-evaluations of the cost contribution of each player to the grand coalition, subject to the core constraints when the core of the game is not empty. Since the vehicle-routing game may have an empty core, an extended core concept is also proposed for the game, and the cross-evaluation value is then applied to its cost allocation. Numerical examples are presented and discussed