Empirical Evidence of an Efficient Formulation for the Multi-period Setup Carryover Lot Sizing Problem (original) (raw)
Related papers
2010
We investigate different formulations of the multi item, multi period capacitated lot sizing problem with inclusions of backorders, setup times and setup costs into it. The problem is closer to the realistic situations and is abbreviated as CLSP_BS in this work. Apart from the classical formulation, we give two variants of the transportation formulation of CLSP_BS. Objective values of these three formulations are exactly equivalent to each other, but they rank different in terms of computational times. When we compare the bounds obtained by LP relaxation of the classical and the two transportation formulations, it is observed that classical and one of the two transportation formulations are exactly equivalent; however the other transportation formulation generates a comparatively better bound. Based on this information on strength of bounds, we earmark the formulations of CLSP_BS as strong and weak. This knowledge about strong and weak formulations can prove to be fruitful while solving real life large sized problems. Limited computational experiences are shown here which establish the stated claims.
2010
In this work we introduce an innovative procedure to solve the capacitated lot sizing problem with backorders and setup times, called CLSP_BS. The procedure formulates CLSP_BS as a mixed integer programming (MIP) problem, reduces it to the structure of a bounded variable linear program; and then calculates some ratios of the coefficients to determine an approximate solution to the problem. This initial solution is further improved using an intelligent enumeration procedure. Adopting this procedure, we solve the NP hard MIP differently and easily, by mere calculation of a few ratios and without actually using any traditional solution approaches, viz. simplex, interior point method, etc.
Multi-item lot-sizing with joint set-up costs
Mathematical Programming, 2009
We consider a multi-item lot-sizing problem with joint set-up costs and constant capacities. Apart from the usual per unit production and storage costs for each item, a set-up cost is incurred for each batch of production, where a batch consists of up to C units of any mix of the items. In addition, an upper bound on the number of batches may be imposed.
Models for capacitated lot-sizing problem with backlogging, setup carryover and crossover
Journal of the Operational Research Society, 2013
Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.
Multi-Item Capacity Constrained Dynamic Lot-Sizing and Sequencing with Setup Time
Journal of Mechanical Engineering, 2010
Production lot-sizing has a special significance in supply chain taking into account the fact that majority of the lot-sizing problems are associated with NP-hard scheduling and sequencing problems. The complexity increases exponentially when multi-item capacitated dynamic lot-sizing is considered. The basic economic production quantity (EPQ) model minimizes the sum of setup and holding cost under certain favorable assumptions. However, when assumptions are removed by introducing more complex constraints, the solution procedure becomes extremely difficult to solve. As a result NP-hardness arises which necessitates the use of heuristics. The objective of this paper is to minimize the sum of setup and inventory holding costs over a time horizon subject to constraints of capacity limitations and elimination of backlogging. As reports reveal, algorithm for an optimal solution exists in case of a single item production. But for multi-item problems, no algorithm exists which can provide g...
Journal of the Operational Research Society, 2006
We address the multi-item, capacitated lot-sizing problem (CLSP) encountered in environments where demand is dynamic and to be met on time. Items compete for a limited capacity resource, which requires a setup for each lot of items to be produced causing unproductive time but no direct costs. The problem belongs to a class of problems that are difficult to solve. Even the feasibility problem becomes combinatorial when setup times are considered. This difficulty in reaching optimality and the practical relevance of CLSP make it important to design and analyse heuristics to find good solutions that can be implemented in practice. We consider certain mixed integer programming formulations of the problem and develop heuristics including a curtailed branch and bound, for rounding the setup variables in the LP solution of the tighter formulations. We report our computational results for a class of instances taken from literature.
Hybrid Formulation of the Multi-Item Capacitated Dynamic Lot Sizing Problem
American Journal of Operations Research, 2015
It is shown that when backorders, setup times and dynamic demand are included in capacitated lot sizing problem, the resulting classical formulation and one of the transportation formulations of the problem (referred to as CLSP_BS) are equivalent. And it is shown that both the formulations are "weak" formulations (as opposed to "strong" formulation). The other transportation version is a strong formulation of CLSP_BS. Extensive computational studies are presented for medium and large sized problems. In case of medium-sized problems, strong formulation produces better LP bounds, and takes lesser number of branch-and-bound (B&B) nodes and less CPU time to solve the problem optimally. However for large-sized problems strong formulation takes more time to solve the problem optimally, defeating the benefit of strength of bounds. This essentially is because of excessive increase in the number of constraints for the large sized problems. Hybrid formulations are proposed where only few most promising strong constraints are added to the weak formulation. Hybrid formulation emerges as the best performer against the strong and weak formulations. This concept of hybrid formulation can efficiently solve a variety of complex real life large-sized problems.
Multi-item Lot-sizing with a Joint Set-up Cost
SSRN Electronic Journal, 2000
We consider a multi-item lot-sizing problem in which there are demands, and unit production and storage costs. In addition production of any mix of items is measured in batches of fixed size, and there is a fixed set-up cost per batch in each period. Suppose that the unit production costs are constant over time, the storage costs are nonnegative, and for any two items the one that has a higher storage cost in one period has a higher storage cost in every period. Then we show that there is a linear program with O(mT 2 ) constraints and variables that solves the multi-item lot-sizing problem, thereby establishing that it is polynomially solvable, where m is the number of items and T the number of time periods. This generalizes an earlier result of Anily and Tzur who presented a O(mT m+5 ) dynamic programming algorithm for essentially the same problem. Under additional conditions, a similar linear programming result is shown to hold in the presence of backlogging when the batch size is arbitrarily large.
The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs
Computers & Operations Research, 2009
We address a multi-item capacitated lot-sizing problem with setup times, safety stock and demand shortages. Demand cannot be backlogged, but can be totally or partially lost. Safety stock is an objective to reach rather than an industrial constraint to respect. The problem is np-hard. We propose a Lagrangian relaxation of the resource capacity constraints. We develop a dynamic programming algorithm to solve the induced sub-problems. An upper bound is also proposed using a Lagrangian heuristic with several smoothing algorithms. Some experimental results showing the effectiveness of the approach are reported.
This paper presents a novel mathematical programming approach to the singlemachine capacitated lot-sizing and scheduling problem with sequence-dependent setup times and setup costs. The approach is partly based on the earlier work of Haase and Kimms (2000) which determines during pre-processing all item sequences that can appear in given time periods in optimal solutions. We introduce a new mixed-integer programming model in which binary variables indicate whether individual items are produced in a period, and parameters for this program are generated by a heuristic procedure in order to establish a tight formulation. Our model allows us to solve in reasonable time instances where the product of the number of items and number of time periods is at most 60-70. Compared to known optimal solution methods, it solves significantly larger problems, often with orders of magnitude speedup.