Mohamed Ali Aloulou

Complexity of single machine scheduling problems under scenario-based uncertainty

We present algorithmic and computational complexity results for several single machine scheduling problems where some job characteristics are uncertain. This uncertainty is modeled through a finite set of well-defined scenarios. We use here the so-called absolute robustness criterion to select among feasible solutions.

Coordination of production and interstage batch delivery with outsourced distribution

In this paper, we consider coordinated production and interstage batch delivery scheduling problems, where a third-party logistics provider (3PP) delivers semi-finished products in batches from one production location to another production location belonging to the same manufacturer. A batch cannot be delivered until all jobs of the batch are completed at the upstream stage. The 3PP is required to deliver each product within a time T from its release at the upstream stage. We consider two transportation modes: regular transportation, for which delivery departure times are fixed at the beginning, and express transportation, for which delivery departure times are flexible. We analyze the problems faced by the 3PP when either the manufacturer dominates or the 3PP dominates. In this context, we investigate the complexity of several problems, providing polynomiality and NP-completeness results.

A bibliography of non-deterministic lot-sizing models

Non-deterministic lot-sizing models are considered which serve for an explicit determination of lot sizes in an uncertain environment. Taxonomy components for such models are suggested and a bibliography structured according to these components is presented. The taxonomy components are numeric characteristics of a lot-sizing problem, names of uncertain parameters and names of approaches to model the uncertainty. The bibliography covers more than 300 publications since the year 2000.

Minimizing the number of late jobs on a single machine under due date uncertainty

We study the problem of minimizing the number of late jobs on a single machine where job processing times are known precisely and due dates are uncertain. The uncertainty is captured through a set of scenarios. In this environment, an appropriate criterion to select a schedule is to find one with the best worst-case performance, which minimizes the maximum number of late jobs over all scenarios. For a variable number of scenarios and two distinct due dates over all scenarios, the problem is proved NP-hard in the strong sense and non-approximable in pseudo-polynomial time with approximation ratio less than 2. It is polynomially solvable if the number s of scenarios and the number v of distinct due dates over all scenarios are given constants. An O(nlog n) time s-approximation algorithm is suggested for the general case, where n is the number of jobs, and a polynomial 3-approximation algorithm is suggested for the case of unit-time jobs and a constant number of scenarios. Furthermore, an O(ns+v−2/(v−1)v−2) time dynamic programming algorithm is presented for the case of unit-time jobs. The problem with unit-time jobs and the number of late jobs not exceeding a given constant value is solvable in polynomial time by an enumeration algorithm. The obtained results are related to a min-max assignment problem, an exact assignment problem and a multi-agent scheduling problem.

Two faster algorithms for coordination of production and batch delivery: A note

This note suggests faster algorithms for two integrated production/distribution problems studied earlier, improving their complexities from O(n2V + 4) and O(n2(L + V)2) to O(n) and O(n + Vmin {V, n}) respectively, where n is the number of products to be delivered, V is the number of vehicles and L is the number of vehicle departure times.

Flexible solutions in disjunctive scheduling: General formulation and study of the flow-shop case

We consider the context of decision support for schedule modification after the computation off-line of a predictive optimal (or near optimal) schedule. The purpose of this work is to provide the decision maker a characterization of possible modifications of the predictive schedule while preserving optimality. In the context of machine scheduling, the anticipated modifications are changes in the predictive order of operations on the machines. To achieve this goal, a flexible solution feasible w.r.t to operations deadlines, is provided instead of a single predictive schedule. A flexible solution represents a set of semi-active schedules and is characterized by a partial order on each machine, so that the total order can be set on-line, as required by the decision maker. A flexible solution is feasible if all the complete schedules that can be obtained by extension are also feasible. In this paper we develop two main issues. The first one concerns the evaluation of a flexible solution in the worst case allowing to certify if the solution is feasible. The second issue is the computation of feasible (w.r.t deadlines) flexible solutions of maximal flexibility imposed by the decision maker. Under an epsilon-constraint framework, solving this problem allows to find compromise solutions for the flexibility criterion and any minmax regular scheduling criterion. The special case of the flow-shop scheduling problem is studied and computational experiments are carried out.

Integrated production scheduling and vehicle routing problem with job splitting and delivery time windows

Abstract In this paper, we study a production scheduling and vehicle routing problem with job splitting and delivery time windows in a company working in the metal packaging industry. In this problem, a set of jobs has to be processed on unrelated parallel machines with job splitting and sequence-dependent setup time (cost). Then the finished products are delivered in batches to several customers with heterogeneous vehicles, subject to delivery time windows. The objective of production is to minimize the total setup cost and the objective of distribution is to minimize the transportation cost. We propose mathematical models for decentralized scheduling problems, where a production schedule and a distribution plan are built consecutively. We develop a two-phase iterative heuristic to solve the integrated scheduling problem. We evaluate the benefits of coordination through numerical experiments.

Evaluating flexible solutions in single machine scheduling via objective function maximization : The study of computational complexity

We study a deterministic problem of evaluating the worst case performance of flexible solutions in the single machine scheduling. A flexible solution is a set of schedules following a given structure determined by a partial order of jobs and a type of the schedules. In this paper, the schedules of active and non-delay type are considered. A flexible solution can be used on-line to absorb the impact of data disturbances related to, for example, job arrival, tool availability or machine breakdowns. The performance of a flexible solution includes the best case and the worst case performances. The best case performance is an ideal performance that can be achieved only if the on-line conditions allow to implement the best schedule of the set of schedules characterizing the flexible solution. In contrast, the worst case performance indicates how poorly the flexible solution may perform when following the given structure in the on-line circumstances. The best-case and the worst-case performances are usually evaluated by the minimum and maximum values of the considered objective function, respectively. We present algorithmic and computational complexity results for some maximization scheduling problems. In these problems, the jobs to be scheduled have different release dates and precedence constraints may be given on the set of jobs.

Worst-case evaluation of flexible solutions in disjunctive scheduling problems

In this paper, we consider the problem of evaluating the worst case performance of flexible solutions in non-preemptive disjunctive scheduling. A flexible solution represents a set of semi-active schedules and is characterized by a partial order on each machine. A flexible solution can be used on-line to absorb the impact of some data disturbances related for example to job arrival, tool availability and machine breakdowns. Providing a flexible solution is useful in practice only if it can be assorted with an evaluation of the complete schedules that can be obtained by extension. For this purpose, we suggest to use only the best case and the worst case performance. The best case performance is an ideal performance that can be achieved only if the on-line conditions allow to implement the best schedule among the set of schedules characterized by the flexible solution. In contrast, the worst case performance indicates how poorly the flexible solution may perform. These performances can be obtained by solving corresponding minimization and maximization problems. We focus here on maximization problems when a regular min-max objective function is considered. In this case, the worse objective function value can be determined by computing the worse completion time of each operation separately. We show that this problem can be solved by finding an elementary longest path in the disjunctive graph representing the problem with additional constraints. In the special case of the flow-shop problem with release dates and additional precedence constraints, we give a polynomial algorithm that determines the worst case performance of a flexible solution.

A bicriteria two-machine flow-shop serial-batching scheduling problem with bounded batch size

We consider the two-machine flow-shop serial-batching scheduling problem where the machines have a limited capacity in terms of the number of jobs. Two criteria are considered here. The first criterion is the number of batches to be minimized. This criterion reflects situations where processing of any batch induces a fixed cost, which leads to a total cost proportional to the number of batches. The second criterion is the makespan. This model is relevant in different production contexts, especially when considering joint production and inbound delivery scheduling. We study the complexity of the problem and propose two polynomial-time approximation algorithms with a guaranteed performance. The effectiveness of these algorithms is evaluated using numerical experiments. Exact polynomial-time algorithms are also provided for some particular cases.