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

Stability analysis of interval type-2 sampled-data polynomial fuzzy-model-based control system with a switching control scheme

The stability of an interval type-2 (IT2) sampled-data (SD) polynomial fuzzy-model-based control system with a switching control scheme is studied in this paper. The uncertain nonlinear plant is depicted via an IT2 polynomial fuzzy model. To realize control, a switching IT2SD polynomial fuzzy controller is generated. This paper adopts a switching control scheme with a variable sampling period. The modeling domain consists of several sub-domains, and each sub-domain corresponds to a local IT2SD polynomial fuzzy controller. These local IT2SD polynomial fuzzy controllers form the switching IT2SD polynomial fuzzy controller. To aid in the stability analysis, this paper adopts a looped-functional-based technique. The imperfect premise matching concept is brought in to solve the mismatch dilemma caused by the SD control strategy and uncertainties. For decreasing the conservativeness, this paper takes into account the state information as well as the information of IT2 membership functions. The stability analysis is performed for each sub-domain, providing the potential for further relaxation. As polynomials exist in the stability conditions, this paper employs the sum-of-squares method for the stability investigation. The simulation outcomes confirm the efficacy of the proposed method

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

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

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