Nadjib Brahimi

Single item lot sizing problems

A state-of-the-art of a particular planning problem, the Single Item Lot Sizing Problem (SILSP), is given for its uncapacitated and capacitated versions. First classes of lot sizing problems are briefly surveyed. Various solution methods for the Uncapacitated Single Item Lot Sizing Problem (USILSP) are reviewed. Four different mathematical programming formulations of the classical problem are presented. Different extensions for real-world applications of this problem are discussed. Complexity results of the Capacitated Single Item Lot Sizing Problem (CSILSP) are given together with its different formulations and solution techniques.

Capacitated Multi-Item Lot-Sizing Problems with Time Windows

This research concerns a new family of capacitated multi-item lot-sizing problems, namely, lot-sizing problems with time windows. Two classes of the problem are analyzed and solved using different Lagrangian heuristics. Capacity constraints and a subset of time window constraints are relaxed resulting in particular single-item time window problems that are solved in polynomial time. Other relaxations leading to the classical Wagner-Whitin problem are also tested. Several smoothing heuristics are implemented and tested, and their results are compared. The gaps between lower and upper bounds for most problems are very small (less than 1%). Moreover, the proposed algorithms are robust and do not seem to be too affected when different parameters of the problem are varied.

Integrated production planning and order acceptance under uncertainty: A robust optimization approach

The aim of this paper is to formulate a model that integrates production planning and order acceptance decisions while taking into account demand uncertainty and capturing the effects of congestion. Orders/customers are classified into classes based on their marginal revenue and their level of variability in order quantity (demand variance). The proposed integrated model provides the flexibility to decide on the fraction of demand to be satisfied from each customer class, giving the planner the choice of selecting among the highly profitable yet risky orders or less profitable but possibly more stable orders. Furthermore, when the production stage exceeds a critical utilization level, it suffers the consequences of congestion via elongated lead-times which results in backorders and erodes the firm’s revenue. Through order acceptance decisions, the planner can maintain a reasonable level of utilization and hence avoid increasing delays in production lead times. A robust optimization (RO) approach is adapted to model demand uncertainty and non-linear clearing functions characterize the relationship between throughput and workload to reflect the effects of congestion on production lead times. Illustrative simulation and numerical experiments show characteristics of the integrated model, the effects of congestion and variability, and the value of integrating production planning and order acceptance decisions.

Polyhedral and Lagrangian approaches for lot sizing with production time windows and setup times

In this paper, we solve the capacitated multi item lot-sizing problem with non-customer specific production time windows and setup times using two approaches: (i) using a Lagrangian relaxation-based heuristic and (ii) using reformulations and a commercial software. The results of the two approaches are analyzed and compared based on randomly generated data sets. The results show that the first approach finds feasible solution more rapidly but a steady state is reached very quickly. On the other hand the second approach quickly finds good lower bounds and finds good feasible solutions if more CPU time is allowed. It turns out that, for a wide variety of instances varying in size and other parameters, we can obtain feasible solutions within 1-5% of optimal within 10s and also obtain solutions that are guaranteed within 1-2% of optimal within 60-120s.

Single-item dynamic lot-sizing problems: An updated survey

Following our previous paper (Brahimi, Dauzere-Peres, Najid, & Nordli, 2006), we present an updated and extended survey of Single-Item Lot-Sizing Problems with focus on publications from 2004 to 2016. Exact and heuristic solution procedures are surveyed. A concise and comprehensive summary of different extensions of the problem is given. The classification of the extensions is based on different characteristics such as resource limitations, assumptions on demand and cost structure. The large number of surveyed papers shows the increased interest of researchers in lot-sizing problems in general and in single-item problems in particular. The survey and the proposed classification should help researchers to identify new research topics, to propose relevant problems and/or novel solution approaches.

Multi-item production routing problem with backordering: a MILP approach

The aim of this paper is to present mixed integer linear programming formulations for the production routing problem with backordering (PRP-B) and a new hybrid heuristic to solve the problem. The PRP-B is considered in the context of a supply chain consisting of a production facility with limited production and storage capacities and geographically dispersed points of sale with limited storage capacities. The PRP-B integrates multiple item lot sizing decisions and vehicle routing decisions to the points of sale, where backordering of end customer demands is allowed at a penalty. Two integrated mixed integer programming models are formulated and a solution procedure consisting of a relax-and-fix heuristic combined with a local search algorithm is proposed. The numerical results show that this hybrid heuristic outperforms a state-of-the-art MIP commercial solver, in terms of solution quality and CPU times.

Integrating order acceptance decisions with flexible due dates in a production planning model with load-dependent lead times

We consider a tactical planning problem, which integrates production planning decisions together with order acceptance decisions, while taking into account the dependency between workload and lead times. The proposed model determines which orders to accept and in which period they should be produced, so that they can be delivered to the customer within the acceptable flexible due dates. When the number of accepted orders increases, the workload and production lead time also increase, and this may result in the possibility of missing customer due dates. This problem is formulated as a mixed integer linear programme for which two relax-and-fix heuristic solution methods are proposed. The first one decomposes the problem based on time periods, while the second decomposes it based on orders. The performances of these heuristics are compared with that of a state-of-the-art commercial solver. Our results show that the time-based relax-and-fix heuristic outperforms the order-based relax-and-fix heuristic, and the solver solution as it yields better integrality gaps for much less CPU effort.

Models and Lagrangian heuristics for a two-level lot-sizing problem with bounded inventory

We consider a two-level dynamic lot-sizing problem where the first level consists of N finished products competing for a single type of purchased raw material in the second level. While the procurement and production capacities are unlimited, the storage capacity of the raw material is limited and must be carefully managed. The goal is to simultaneously determine a replenishment plan for the raw material and optimal production plans for the finished products on a horizon of T periods while minimizing production, purchasing, setup and inventory holding costs. The problem is modeled using mixed-integer linear programs and solved using both a Lagrangian relaxation-based heuristic and a commercial mixed-integer linear programming solver. Learning capabilities are integrated in the Lagrangian relaxation to update step size in the subgradient algorithm. The computational results show that the Lagrangian heuristic outperforms the solver on different formulations, in particular for large problems with long time horizons.

Production planning with order acceptance and demand uncertainty

Traditional production planning models assume that all orders must be satisfied when capacity is available. In this paper, we analyze the value of providing decision makers with the flexibility to accept or reject orders, when order quantity is uncertain. We introduce this demand flexibility in two production planning problems. The first problem integrates order acceptance in the capacitated lot sizing problem, providing the option to reject an order if it requires a high setup cost and cannot be aggregated with additional orders to take advantage of economies of scale. The second problem integrates order acceptance in the order release planning problem with load-dependent lead times (LDLTs). This problem provides the option to reject an order if it increases the workload causing the delay of other orders due to congestion effects. Robust counterparts of both integrated problems are formulated as linear mixed integer programs (MIPs). The deterministic integrated problems and their robust counterparts are shown to be NP-hard and a two-stage MIP heuristic is proposed as a solution procedure. A relax and fix (RF) heuristic is adapted to efficiently construct feasible solutions to the robust problems, which are then improved by a fix and optimize (FO) heuristic. Numerical results show that the proposed heuristics give promising results in terms of solution quality and computation time. Simulation experiments are conducted to assess the value of demand flexibility and to study the effects of various parameters on economical performance.

Integrated dynamic single item lot-sizing and quality inspection planning

This paper proposes an integrated model for single item dynamic lot-sizing problem and Quality Inspection Planning (QIP). The objective is to provide a model of production planning that takes into account a targeted level of outgoing quality or an Acceptable quality level (AQL) when the manufacturing system inherently generates a proportion of defectives that increases significantly when the system switches from the in-control state to the out-of-control state. The average outgoing quality of each period of time of the planning horizon is bounded as a function of the inspection capacity. The effects of integrating QIP are analysed and discussed through several experiments representing different quality control system’s parameters, i.e. inspection capacity, inspection cost and AQL. The simulation results show that it is very important to take into account the inspection process into production planning decisions. This study will help the decision-makers to negotiate service levels or react properly to given customer quality requirements based on cost and lead time parameters in addition to their process characteristics in terms of capability and stability.