Moustapha Diaby

Capacitated lot-sizing and scheduling by Lagrangean relaxation

Abstract Development of new models and solution procedures for production planning has been of research interest for several decades. Implementation of these models has resulted in lower production costs by reducing inventories, number of setups and labor costs. In this paper, we develop several optimal/near-optimal procedures for the Capacitated Lot-Sizing and Scheduling Problem (CLSP) with setup times, limited regular time and limited overtime. We formulate a mixed-integer linear programming model of the problem and solve it by Lagrangean relaxation. We experiment with alternative Lagrangean relaxations and develop new procedures to solve these relaxations. Overall, the capacity constraints relaxation seems to be superior to the demand constraints relaxation. Our results show that large problems can be solved in reasonable computer times and within one-percent accuracy of the optimal solutions. We solved 99 × 8 (i.e., 99 items and 8 periods), 50 × 12 and 50 × 8 problems in 30.61, 36.25 and 12.65 seconds of CDC Cyber 730 computer time, respectively. Our procedures are general enough to be applied directly or with slight modifications in real-life production settings.

Dynamic lot sizing for multi-echelon distribution systems with purchasing and transportation price discounts

We consider the problem of determining optimal purchasing and shipping quantities over a finite planning horizon for arborescent, multi-echelon physical distribution systems with deterministic, time-varying demands. We assume that the inventory holding cost at a given warehouse of the distribution network is a linear function of the inventory level, and that the total procurement cost (i.e., ordering, plus purchasing, plus transportation and reception costs) is a general piecewise-linear function of the quantities shipped to and from the warehouse. We formulate a mixed integer linear programming model of the problem and develop a Lagrangian relaxation-based procedure to solve it. We show computational results for problems with 12 periods, up to 15 warehouses, and 3 transportation price ranges.

Multiple items procurement under stochastic nonstationary demands

Abstract In the consumer goods wholesaling and retailing industry, a large number of stock keeping units must be managed on a regular basis. The items are typically purchased in families, each family being associated with a specific external vendor, and usually there are some constraints on family order quantities and there are frequent opportunities to buy at a temporary low price. The demand for an item is often stochastic and not stationary. In this paper, by computing procurement plans over rolling planning horizons, we transform this difficult sequential decision problem into a multi-period static decision problem under risk. The problem is initially formulated as a stochastic program with simple recourse and a branch and bound algorithm is designed to solve an equivalent deterministic program. A piecewise concave approximation is proposed to reduce this program to a linear program with one 0–1 variable per planning period. The performances of the algorithms are studied in two simulation experiments. The simulations show that, when planning over a rolling horizon, the approach proposed yields excellent results.

Implicit Enumeration for the Pure Integer 0/1 Minimax Programming Problem

We present an implicit enumeration procedure for solving pure integer 0/1 minimax problems which arise in the context of Benders decomposition for mixed integer 0/1 linear programming problems, or in various practical settings such as the location of facilities and assembly line balancing. The procedure is an extension of the additive algorithm of E. Balas for pure integer 0/1 programming problems. We solve minimax problems directly i.e., as minimax problems, not as mixed integer programming problems. A numerical example is used to illustrate the procedure. Extensions of the basic algorithm are discussed.

Efficient post-optimization analysis procedure for the dynamic lot-sizing problem

Abstract In this paper, we develop an efficient post-optimization analysis procedure for the Wagner-Whitin solution to the Dynamic Lot-Sizing Problem (DLSP). The proposed procedure can be used in the context of a branch-and-bound algorithm, or in smoothing heuristic approaches for solving the Capacitated Lot-Sizing Problem (CLSP).