Capacitated dynamic lot sizing problems in closed-loop supply chain

In this paper, we address the capacitated dynamic lot sizing problem arising in closed-loop supply chain where returned products are collected from customers. These returned products can either be disposed or be remanufactured to be sold as new ones again; hence the market demands can be satisfied by either newly produced products or remanufactured ones. The capacities of production, disposal and remanufacturing are limited, and backlogging is not allowed. A general model of this problem is formulated, and several useful properties of the problem are characterized when cost functions are concave. Moreover, this problem is analyzed and solved to optimality using dynamic programming algorithms under different scenarios. It is shown that the problem with only disposal or remanufacturing can be converted into a traditional capacitated lot sizing problem and be solved by a polynomial algorithm if the capacities are constant. A pseudo-polynomial algorithm is proposed for the problem with both capacitated disposal and remanufacturing. The problem with capacitated production and remanufacturing and the problem with uncapacitated production and capacitated remanufacturing are also analyzed and solved. Through numerical experiments we show that the proposed algorithms perform well when solving problems of practical sizes. From the experimental results also indicates that it is worthwhile to expand the remanufacturing capacity only when returned products exist in a relatively long planning horizon, and production capacities have little effect on the remanufacturing plan when the demand is mainly satisfied by the production.     

Capacitated lot sizing and scheduling with parallel machines and shared buffers: A case study in a packaging company

The aim of this work is to propose a solution approach for a capacitated lot sizing and scheduling real problem with parallel machines and shared buffers, arising in a packaging company producing yoghurt. The problem has been formulated as a hybrid Continuous Set-up and Capacitated Lot Sizing Problem (CSLP–CLSP). A new effective two stage optimisation heuristic based on the decomposition of the problem into a lot sizing problem and a scheduling problem has been developed. An assignment of mixture to buffers is made in the first stage, and therefore the corresponding orders are scheduled on the production lines by performing a local search. Computational tests have been performed on the real data provided by the company. The heuristic exhibits near-optimal solutions, all obtained in a very short computational time.     

GRASP heuristic with path-relinking for the multi-plant capacitated lot sizing problem

This paper addresses the independent multi-plant, multi-period, and multi-item capacitated lot sizing problem where transfers between the plants are allowed. This is an NP-hard combinatorial optimization problem and few solution methods have been proposed to solve it. We develop a GRASP (Greedy Randomized Adaptive Search Procedure) heuristic as well as a path-relinking intensification procedure to find cost-effective solutions for this problem. In addition, the proposed heuristics is used to solve some instances of the capacitated lot sizing problem with parallel machines. The results of the computational tests show that the proposed heuristics outperform other heuristics previously described in the literature. The results are confirmed by statistical tests.     

Pricing,relaxing and fixing under lot sizing and scheduling

We present a novel mathematical model and a mathematical programming based approach to deliver superior quality solutions for the single machine capacitated lot sizing and scheduling problem with sequence-dependent setup times and costs. The formulation explores the idea of scheduling products based on the selection of known production sequences. The model is the basis of a matheuristic, which embeds pricing principles within construction and improvement MIP-based heuristics. A partial exploration of distinct neighborhood structures avoids local entrapment and is conducted on a rule-based neighbor selection principle. We compare the performance of this approach to other heuristics proposed in the literature. The computational study carried out on different sets of benchmark instances shows the ability of the matheuristic to cope with several model extensions while maintaining a very effective search. Although the techniques described were developed in the context of the problem studied, the method is applicable to other lot sizing problems or even to problems outside this domain.     

Hybrid heuristics for the capacitated lot sizing and loading problem with setup times and overtime decisions

The capacitated lot sizing and loading problem (CLSLP) deals with the issue of determining the lot sizes of product families/end items and loading them on parallel facilities to satisfy dynamic demand over a given planning horizon. The capacity restrictions in the CLSLP are imposed by constraints specific to the production environment considered. When a lot size is positive in a specific period, it is loaded on a facility without exceeding the sum of the regular and overtime capacity limits. Each family may have a different process time on each facility and furthermore, it may be technologically feasible to load a family only on a subset of existing facilities. So, in the most general case, the loading problem may involve unrelated parallel facilities of different classes. Once loaded on a facility, a family may consume capacity during setup time. Inventory holding and overtime costs are minimized in the objective function. Setup costs can be included if setups incur costs other than lost production capacity. The CLSLP is relevant in many industrial applications and may be generalized to multi-stage production planning and loading models. The CLSLP is a synthesis of three different planning and loading problems, i.e., the capacitated lot sizing problem (CLSP) with overtime decisions and setup times, minimizing total tardiness on unrelated parallel processors, and, the class scheduling problem, each of which is NP in the feasibility and optimality problems. Consequently, we develop hybrid heuristics involving powerful search techniques such as simulated annealing (SA), tabu search (TS) and genetic algorithms (GA) to deal with the CLSLP. Results are compared with optimal solutions for 108 randomly generated small test problems. The procedures developed here are also compared against each other in 36 larger size problems.     

A multi-class multi-level capacitated lot sizing model

When demand loading is higher than available capacity, it takes a great deal of effort for a traditional MRP system to obtain a capacity-feasible production plan. Also, the separation of lot sizing decisions and capacity requirement planning makes the setup decisions more difficult. In a practical application, a production planning system should prioritize demands when allocating manufacturing resources. This study proposes a planning model that integrates all MRP computation modules. The model not only includes multi-level capacitated lot sizing problems but also considers multiple demand classes. Each demand class corresponds to a mixed integer programming (MIP) problem. By sequentially solving the MIP problems according to their demand class priorities, this proposed approach allocates finite manufacturing resources and generates feasible production plans. In this paper we experiment with three heuristic search algorithms: (1) tabu search; (2) simulated annealing, and (3) genetic algorithm, to solve the MIP problems. Experimental designs and statistical methods are used to evaluate and analyse the performance of these three algorithms. The results show that tabu search and simulated annealing perform best in the confirmed order demand class and forecast demand class, respectively.     

Improved lower bounds for the capacitated lot sizing problem with setup times

We present new lower bounds for the capacitated lot sizing problem, applying decomposition to the network reformulation. The demand constraints are the linking constraints and the problem decomposes into subproblems per period containing the capacity and setup constraints. Computational results and a comparison to other lower bounds are presented.     

Facets and algorithms for capacitated lot sizing

The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a single-item capacitated version of this problem, and use these results to develop solution methods for multi-item applications. In particular, we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron. Computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane/branch and bound procedure. Moreover, our results show that in many instances adding certain of these inequalities a priori to the problem formulation, and avoiding the generation of cutting planes, can be equally effective.Supported by Grant #ECS-8316224 from the Systems Theory and Operations Research Program of the National Science Foundation.     

Lot sizing with inventory gains

The lot sizing problem with inventory gains generalizes the classical lot sizing problem to one in which stock is not conserved. Instances of this problem can be polynomially transformed into instances of the classical problem. The implications for problems involving different production capacity limitations, backlogging and multilevel production are discussed.     

Planning lot sizes and capacity requirements in a single stage production system

The problem addressed in this paper is the determination of lot sizes for multiple products to be produced on a single production facility with limited capacity. Demand is assumed to be deterministic and time-varying and must be met without backordering. The objective is to minimize the sum of setup and inventory holding costs. A heuristic solution procedure of the period-by-period type is presented. Moreover, the interaction between lot sizing and smoothing of capacity requirements is investigated in a case study.     

Reformulation and a Lagrangian heuristic for lot sizing problem on parallel machines

We consider the capacitated lot sizing problem with multiple items, setup time and unrelated parallel machines. The aim of the article is to develop a Lagrangian heuristic to obtain good solutions to this problem and good lower bounds to certify the quality of solutions. Based on a strong reformulation of the problem as a shortest path problem, the Lagrangian relaxation is applied to the demand constraints (flow constraint) and the relaxed problem is decomposed per period and per machine. The subgradient optimization method is used to update the Lagrangian multipliers. A primal heuristic, based on transfers of production, is designed to generate feasible solutions (upper bounds). Computational results using data from the literature are presented and show that our method is efficient, produces lower bounds of good quality and competitive upper bounds, when compared with the bounds produced by another method from the literature and by high-performance MIP software.     

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lot sizing problem with step-wise production costs (LS-SW). Constant sized batch production is carried out with a limited production capacity in order to satisfy the customer demand over a finite horizon. A new class of valid inequalities we call mixed flow cover, is derived from the existing integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and when V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. We propose a cutting plane algorithm for different classes of valid inequalities introduced in the paper. The exact separation algorithm proposed for the constant capacitated case runs in polynomial time. Computational results show the efficiency of the new class mixed flow cover compared to the existing methods.     

The impact of controllable production rates on the performance of inventory systems: A systematic review of the literature

The lot sizing problem has attracted the attention of researchers for more than a century, and it still belongs to the most relevant decision problems in many manufacturing companies. During the evolution of research on lot sizing, the seminal economic order quantity (EOQ) model proposed by Harris [1913. How many parts to make at once. Factory, the Magazine of Management, 10 (2), 135-136.] has remained the most popular model, despite its limitations. To support lot sizing decisions in practice, researchers have frequently extended Harris’ basic EOQ model to better reflect the characteristics of real production processes. One of these extensions is the consideration of controllable (variable) production rates, which gives production planners more flexibility in managing the build-up and depletion of inventory and in controlling costs.The aim of this paper is to provide a comprehensive and systematic overview of EPQ-type lot sizing models that consider controllable production rates. First, the paper proposes a conceptual framework that captures the characteristics of controllable production rates including the planning horizon (short vs. long term), the number of potential interventions per production run (one vs. multiple), the effect of controllable production rates on the performance of the inventory system (e.g., unit production costs, energy consumption, product quality), and the type of lot sizing model considered (e.g., two-stage models, multi-stage models, multi-item models). Secondly, the paper presents the results of a systematic literature review and evaluates the state-of-research of lot sizing models with controllable production rates. Based on the analysis of the literature, key trends are summarized and promising research opportunities are discussed.     

A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs

This paper deals with the single-item capacitated lot sizing problem with concave production and storage costs, and minimum order quantity (CLSP-MOQ). In this problem, a demand must be satisfied at each period t over a planning horizon of T periods. This demand can be satisfied from the stock or by a production at the same period. When a production is made at period t, the produced quantity must be greater to than a minimum order quantity (L) and lesser than the production capacity (U). To solve this problem optimally, a polynomial time algorithm in O(T⁵) is proposed and it is computationally tested on various instances.     

Lot sizing and scheduling — Survey and extensions

This contribution summarizes recent work in the field of lot sizing and scheduling. The objective is not to give a comprehensive literature survey, but to explain differences of formal models and to provide some first readings recommendations. Our focus is on capacitated, dynamic, and deterministic cases. To underscore the importance of the research efforts, current practice is described and its shortcomings are exposed. Mathematical programming models where the planning horizon is subdivided into several discrete periods are given for both approaches that are well-established and approaches which may represent tomorrow's state of the art. Two research directions are discussed in more detail: continuous time models and multi-level lot sizing and scheduling. The paper concludes with some advice for future research activities.     

A tabu search heuristic for solving the CLSP with backlogging and set-up carry-over

In this paper, the multi-item, single-level, capacitated, dynamic lot sizing problem with set-up carry-over and backlogging, abbreviated to CLSP⁺, is considered. The problem is formulated as a mixed integer programming problem. A heuristic method consisting of four elements: (1) a demand shifting rule, (2) lot size determination rules, (3) checking feasibility conditions and (4) set-up carry-over determination, provides us with an initial feasible solution. The resulting feasible solution is improved by adopting the corresponding set-up and set-up carry-over schedule and re-optimizing it by solving a minimum-cost network flow problem. Then the improved solution is used as a starting solution for a tabu search procedure, with the value of moves assessed using the same minimum-cost network problem. Computational results on randomly generated problems show that the algorithm, which is coded in C++, is able to provide optimal solutions or solutions extremely close to optimal. The computational efficiency makes it possible to solve reasonably large problem instances routinely on a personal computer.     

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called 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 decomposition-based 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.     

考虑机器调整次数和产品质量的卷烟批量计划和柔性流水车间调度集成问题

针对卷烟企业生产中的批量计划和柔性流水车间调度集成问题,构建了整数规划模型,目标函数由卷烟生产时间、生产线调整次数、卷烟质量、库存成本四部分组成。鉴于该问题的NP-hard性,设计遗传算法进行求解,通过合理设计遗传算子,避免不可行解出现。应用某卷烟企业数据得到优化排产结果,与该企业之前依照经验排产方案进行对比,发现优化排程结果在减少品牌转换次数,提高生产的连续性方面具有明显优势。该算法已作为某卷烟企业排产人员的排产参考,应用于排产决策中,取得了良好的效果,对卷烟企业制定排产计划具有一定的实际指导意义。     

Comparison of bundle and classical column generation

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm. This research has been supported by Inria New Investigation Grant “Convex Optimization and Dantzig-Wolfe Decomposition”.     

Hybrid heuristics for the multi-stage capacitated lot sizing and loading problem

The multi-stage capacitated lot sizing and loading problem (MCLSLP) deals with the issue of determining the lot sizes of product items in serially-arranged manufacturing stages and loading them on parallel facilities in each stage to satisfy dynamic demand over a given planning horizon. It is assumed that regular time capacity decisions have already been made in the tactical level and allocated to the stages, but it is still an important decision problem whether to augment regular time capacity by overtime capacity. Each item may be processed on a technologically feasible subset of existing facilities with different process and setup times on each facility. Since the solution of the MCLSLP requires the design of a powerful algorithm, simulated annealing (SA) and genetic algorithms (GA) are integrated to enhance their individual performances. Furthermore, these global optimisation methods are incorporated into a Lagrangean relaxation scheme, hence creating a hybrid solution methodology. Numerical results obtained using these methods confirm the mutual benefits of integrating different solution techniques.