Multi-item lot-sizing with joint set-up costs (original) (raw)
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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.
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.
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...
IFIP Advances in Information and Communication Technology, 2013
In this paper we present an effective flexible formulation for the capacitated multi-item lot-sizing problem with setup carryovers and setup times. The formulation can accommodate setup times, single or multi-period setup carry-overs, backorders with limits on the number of backorder periods, and shelf-life restrictions without the need for any additional variables and constraints. We provide empirical evidence of the superiority of our model over conventional formulations by comparing LP lower bounds generated on a number of randomly generated test problems. Our flexible formulation dominated the results in 100% of the problem cases.
An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging
European Journal of Operational Research, 2011
This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions. manuscript no. (Please, provide the mansucript number!) capacitated single level multi-item lot-sizing problem with backlogging. examined the uncapacitated single item lot-sizing problem with backlogging and start-up costs, when Wagner-Whitin costs are assumed. Cheng et al. (2001) formulated single-level lot-sizing problems with provisions for backorders using a fixed-charge transportation model and proposed a heuristic solution method. Ganas and Papachristos (2005) proposed a polynomial-time algorithm for the single-item lot-sizing problem with backlogging. Song and Chan (2005) proposed a dynamic programming algorithm for solving a single-item lot-sizing problem with backlogging on a single machine at a finite production rate. Mathieu (2006) proposed two extended linear programming (LP) reformulations of single-item lot-sizing problems with backlogging and constant capacities. In a recent study, Küçükyavuz and Pochet (2009) provided the full description of the convex hull for the single-level uncapacitated problem with backlogging. Wu and Shi (2009b) proposed a heuristic that combines domain knowledge from the different strategies of relax-and-fix effectively for the capacitated multi-level lot sizing problem with the consideration of backlogging. We refer the interested reader to Pochet and Wolsey (2006) for a detailed general review of different lot-sizing problems. We note that the term backlog is used interchangeably with backorder in the lot-sizing literature, referring to any demand that is not satisfied on time but in a later time period, no matter what type of manufacturing environment. In our context, we consider a model that is flexible enough to apply to both MTO (Make-To-Order) and MTS (Make-To-Stock) environments when production is planned based on fixed demands or forecasts. The past research has also considered other classes of lot sizing problems. For example, Thizy and van Wassenhove (1985) designed a Lagrangian relaxation (LR) approach, in which capacity constraints are relaxed, in an attempt to decompose the problem into N uncapacitated single item lot-sizing subproblems, solvable by the Wagner-Whitin algorithm. Trigeiro (1987) developed a similar approach for solving the deterministic, multi-item, single-operation lot-sizing problem. Trigeiro et al. (1989) also proposed LR based methods for large-scale lot-sizing problems. Kuik and Salomon (1990) evaluated a simulated annealing heuristic for solving multi-level lot-sizing problem. Pochet and Wolsey (1991) applied strong cutting planes
A global constraint for the capacitated single-item lot-sizing problem
ArXiv, 2019
The goal of this paper is to set a constraint programming framework to solve lot-sizing problems. More specifically, we consider a single-item lot-sizing problem with time-varying lower and upper bounds for production and inventory. The cost structure includes time-varying holding costs, unitary production costs and setup costs. We establish a new lower bound for this problem by using a subtle time decomposition. We formulate this NP-hard problem as a global constraint and show that bound consistency can be achieved in pseudo-polynomial time and when not including the costs, in polynomial time. We develop filtering rules based on existing dynamic programming algorithms, exploiting the above mentioned time decomposition for difficult instances. In a numerical study, we compare several formulations of the problem: mixed integer linear programming, constraint programming and dynamic programming. We show that our global constraint is able to find solutions, unlike the decomposed constra...
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.
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.
International Journal of Production Economics, 2009
In this paper, we study a special case of the capacitated lot sizing problem (CLSP), where alternative machines are used for the production of a single-item. The production cost on each machine is assumed to be piece-wise linear with discontinuous steps (step-wise costs). The over-produced finished products can be stored in an unlimited storage space to satisfy future demand. The aim is to achieve optimal production planning without backlogging. This problem can be seen as an integration of production and transportation activities in a multi-plant supply chain structure, where finished goods are sent directly from the plants to the distribution center using capacitated vehicles. For this problem, which we show to be NP-hard, we propose an exact pseudo-polynomial dynamic programming algorithm which makes it NP-hard in the ordinary sense. We also give three mixed integer linear programming (MILP) formulations that we have found in the literature for the simplest case of the CLSP. These formulations are adapted to the multi-machine case with a step-wise cost structure, to which some valid inequalities have been added to improve their efficiency. We then compare the computational time of the dynamic program to that of one MILP which we selected among MILP formulations based on its lower computational time and its lower and upper bound quality.
Journal of Global Optimization, 2012
Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the socalled Facility Location formulation, the Shortest Route formulation, and the Inventory and Lot Sizing formulation with ( , S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decompositionbased methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the Relax-and-Fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures.