Economic lot and supply scheduling problem: a time-varying lot sizes approach

We study the economic lot and supply scheduling problem (ELSSP) that arises in the distribution and manufacturing industries. The ELSSP involves the simultaneous scheduling of end-item production and inbound transportation of input materials over an infinite time horizon to minimise the average costs of inventory, production set-up and transportation. We present a new methodology based on a time-varying lot sizes approach for the ELSSP. We also provide computational experiments showing that the developed algorithm outperforms the existing heuristic for improved integrated scheduling.     

Effective allocation of idle time in the group technology economic lot scheduling problem

The group technology economic lot scheduling problem (GT-ELSP) addresses the issue of scheduling the products, classified into groups, on a single facility. The objective is to develop a schedule that minimises the sum of average inventory holding and setup costs over the infinite planning horizon. Previous research on this class of problems has assumed the cycle idle time to be equally divided among various sequence positions and has varied the frequency of production of individual products. We propose a heuristic that maintains constant frequency of products and varies the idle time among the sequence positions. We address the feasibility issue of the GT-ELSP and show that our heuristic, based on the time varying lot size approach, yields low-cost quality solutions that aid practical implementation. We tested our heuristic on a large set of randomly generated problems to assess its efficacy over the solutions of earlier proposed heuristics.     

Accounting for idle capacity cost in the scheduling of economic lot sizes

This paper considers the issue of idle capacity cost in determining economic lot sizes. Two mathematical models are developed for the economic lot scheduling problem (ELSP). In Model I, the ELSP with fixed production rates is formulated under both the common cycle and time-varying lot sizes approaches. The associated constrained optimization problem in the time-varying lot sizes approach is reduced to solving a parametric quadratic programming problem. In Model II, the modified ELSP (or MELSP) is treated with variable production rates and unit production cost of each item as a function of its production rate. An upper bound and a lower bound on the MELSP are derived. Lot-sizing decisions of the proposed models are obtained and their dependencies on the idle capacity cost are examined with numerical examples.     

Stochastic economic lot sizing and scheduling problem with pitch interval,reorder points and flexible sequence

This paper presents a solution for a class of the stochastic economic lot sizing scheduling problem that is typical of the replenishment pull system proposed by the lean manufacturing approach. In this class, lots of any product are produced in fixed intervals called pitch. The proposed solution uses flexible production sequences and reorder points that are compatible with the concepts of supermarket and level production. It adopts the queuing discipline obtained from a fluid model that approximates the stochastic process of arrival and production orders. Given the queuing discipline, an iterative algorithm returns a near-optimal solution for the system. The proposed approach allows us possible to differentiate inventory cost and service levels by product, and the stock required is lower than that required by the discipline ‘first stock out, first out’. The algorithm is fast and stable, allowing its frequent use in real-world instances.     

The economic lot and delivery scheduling problem: common cycle, rework, and variable production rate

The economic lot and delivery scheduling problem is to simultaneously determine the production sequence of several assembly components at a supplier and the delivery interval of those components to the customer. The customer, an assembly facility, is assumed to use the components at a constant rate. The objective is to find the production sequence and delivery interval that minimize the holding, setup, and transportation cost for the supply chain. Previous solutions to the problem assume a constant production rate for each component and that all components are of acceptable quality. These assumptions ignore volume flexibility and quality cost. Volume flexibility permits a system to adjust the production rate upwards or downwards within wide limits. Also, component quality may deteriorate with larger lot sizes and decreased unit production times. In this paper, we develop an algorithm for solving the economic lot and delivery scheduling problem for a supplier using a volume flexible production system where component quality depends on both lot sizes and unit production times. We test the performance of the algorithm and illustrate the models with numerical examples.     

A comparison of solution approaches for the fixed-sequence economic lot scheduling problem

The economic lot scheduling problem (ELSP) has received much attention recently. The general version of the problem has a non-convex objective function, so it is difficult to find truly optimal solutions. We examine the three most popular heuristic approaches to the fixed-sequence ELSP. Each approach imposes one or both of these simplifying constraints: the zero-switch constraint (production of a part is started only when its inventory is depleted) and the equal-lot constraint (the lot size of a given part is constant through time). We provide a formulation that clarifies the relationships between the general problem and the three constrained versions, and compare their performances in a computational study.     

A hybrid genetic algorithm for flowshop lot streaming with setups and variable sublots

Lot streaming is the process of splitting a given lot or job to allow the overlapping of successive operations in flowshops or multi-stage manufacturing systems to reduce manufacturing lead time. Recent literature shows that significant lead time improvement is possible if variable sublots, instead of equal or consistent sublots, are used when production setup time is considered. However, lot streaming problems with variable sublots are difficult to solve to optimality using off-shelf optimisation packages even for problems of small and experimental sizes. Thus, efficient solution procedures are needed for solving such problems for practical applications. In this paper, we develop a mathematical programming model and a hybrid genetic algorithm for solving n-job m-machine lot streaming problems with variable sublots considering setup times. The preliminary computational results are encouraging.     

Hybrid genetic algorithm for group technology economic lot scheduling problem

The concept of group technology has been successfully applied to many production systems, including flexible manufacturing systems. In this paper we apply group technology principles to the economic lot scheduling problem, which has been studied for over 40 years. We develop a heuristic algorithm and a hybrid genetic algorithm for the group technology economic lot scheduling problem. Numerical experiments show that the developed algorithms outperform the existing heuristics.     

Two-stage hybrid flow shop batching and lot streaming with variable sublots and sequence-dependent setups

A paint manufacturing firm's customers typically place orders for two or more products simultaneously. Each product belongs to a family that denotes batching compatibility during manufacturing. Further, products can be split into several sublots to allow overlapping production in a two-stage hybrid flow shop wherein various identical, capacitated machines operate in parallel at each stage. We present a mixed-integer linear program (MILP) for this integrated batching and lot streaming problem with variable sublots, incompatible job families, and sequence-dependent setup times. The model determines the number and size of sublots for each product and the production sequencing for each sublot such that the total weighted completion time is minimised. To promote practical implementation, we develop and evaluate heuristics to efficiently solve this problem.     

Flexible job shop scheduling with lot streaming and sublot size optimisation

Models and optimisation approaches are developed for a flexible job shop scheduling problem with lot streaming and lot sizing of the variable sublots. A two-stage optimisation procedure is proposed. First, the makespan value is minimised with the smallest sublots defined for the problem instance. This makes it possible to shorten the makespan significantly, because each sublot is transferred separately to the next operation of a job. In the second stage, the sizes of the sublots are maximised without increasing the obtained makespan value. In this way, the quantity of sublots and transport activities is limited together with the related manufacturing cost. Two objectives are defined for the second stage. The first one is the maximisation of the sum of the sublot sizes of all operations, the second one is the maximisation of the number of the operations which do not need to be split at all. Mixed-integer linear programming, constraint programming and graph-based models are implemented for the problem. Two optimisation approaches are developed and compared in computational experiments for each stage and objective, one approach is based on a third-party solver, and the second one on an independent own implementation, namely a tabu search and a greedy constructive heuristic.     

On the feasibility of sequence-dependent economic lot scheduling problem

Most of the literature dealing with the determination of cyclic part input sequences in a flexible flow system is restricted in that it only searches for input sequences that are permutations of the minimal part set (MPS). This study is unique in that it investigates input sequences generated by integer programming (IP) formulations that balance or unbalance machine workloads to help maximize overall machine utilization (Stecke 1992). Also, this study integrates the input sequence determination decision with the part mix ratio determination, within the overall framework of a flexible approach to FMS operation over time. A simulation model of a flexible flow system was designed to study the effects on overall machine utilizations caused by utilizing alternative part mix ratios to help determine input sequences. The procedures used to determine the part mix ratios include IP formulations and those that are generated randomly, including the MPS mix ratios. Three different experimental settings are used to test these effects as well as (I) the robustness of the part input sequences that can be derived from the IP generated mix ratios, and (2) the relative importance of the part mix ratio decision in relation to the part input sequence determination. A new FMS physical design that can also be easily modelled to capture look-ahead capability proved to be simple and effective. Several significant conclusions regarding part mix ratios, pan input sequences, and look-ahead capability are revealed. These include: (1) the determination of the part mix ratios proved to be more significant in improving FMS performance than the determination of part input sequences; (2) the robust nature of the IP formulations was demonstrated; and (3) look-ahead capability provides equally high overall machine utilizations at lower levels of work-in-process. Future research areas are presented that would help validate and extend the observations found in this study.     

Aligning workload control theory and practice: lot splitting and operation overlapping issues

This paper addresses the problem of lot splitting in the context of workload control (WLC). Past studies on WLC assumed that jobs released to the shop floor proceed through the different stages of processing without being split. However, in practice, large jobs are often split into smaller transfer sublots so that they can move more quickly and independently through the production process and allow operations overlapping relating to the same job. This paper assesses the performance of different lot splitting policies for job release and dispatching strategies under lot splitting. A new dispatching rule was designed to specifically take advantage of lot splitting and operations overlapping in the context of WLC. Discrete-event simulation is used to assess system performance in relation to the ability to provide shorter delivery times and on time deliveries. Results highlight the importance of releasing the sublots of the same job together and demonstrate that combining an effective lot splitting policy with an appropriate dispatching rule can enhance the performance of production systems.     

An artificial bee colony algorithm for the economic lot scheduling problem

In this study, we present an artificial bee colony (ABC) algorithm for the economic lot scheduling problem modelled through the extended basic period (EBP) approach. We allow both power-of-two (PoT) and non-power-of-two multipliers in the solution representation. We develop mutation strategies to generate neighbouring food sources for the ABC algorithm and these strategies are also used to develop two different variable neighbourhood search algorithms to further enhance the solution quality. Our algorithm maintains both feasible and infeasible solutions in the population through the use of some sophisticated constraint handling methods. Experimental results show that the proposed algorithm succeeds to find the all the best-known EBP solutions for the high utilisation 10-item benchmark problems and improves the best known solutions for two of the six low utilisation 10-item benchmark problems. In addition, we develop a new problem instance with 50 items and run it at different utilisation levels ranging from 50 to 99% to see the effectiveness of the proposed algorithm on large instances. We show that the proposed ABC algorithm with mixed solution representation outperforms the ABC that is restricted only to PoT multipliers at almost all utilisation levels of the large instance.     

The capacitated lot sizing problem with overtime decisions and setup times

The Capacitated Lot Sizing-Problem (CLSP) consists of planning the lot sizes of multiple items over a planning horizon with the objective of minimizing setup and inventory holding costs. In each period that an item is produced a setup cost is incurred. Capacity is limited and homogeneous. Here, the CLSP is extended to include overtime decisions and capacity consuming setups. The objective function consists of minimizing inventory holding and overtime costs. Setups incur costs implicitly via overtime costs, that is, they lead to additional overtime costs when setup times contribute to the use of overtime capacity in a certain period. The resulting problem becomes more complicated than the standard CLSP and requires methods different from the ones proposed for the latter. Consequently, new heuristic approaches are developed to deal with this problem. Among the heuristic approaches are the classical HPP approach and its modifications, an iterative approach omitting binary variables in the model, a Genetic Algorithm approach based on the transportation-like formulation of the single item production planning model with dynamic demand and a Simulated Annealing approach based on shifting family lot sizes among consecutive periods. Computational results demonstrate that the Simulated Annealing approach produces high quality schedules and is computationally most efficient.     

Lot streaming in the presence of learning in sublot-attached setup times and processing times

Lot streaming is the process of splitting a production lot into sublots, and then processing the sublots on different machines in an overlapping manner. In this paper, we study the use of lot streaming for processing a lot in a two-machine flow shop when a sublot-attached setup time is incurred before the processing of each sublot. The objective is to determine number of sublots and sublot sizes and minimize makespan. We also consider the case when the effect of learning is observed in processing times, sublot-attached setup times, or, both. We present closed-form expressions for optimal sublot sizes and efficient search schemes to determine optimal number of sublots.     

An optimal integrated lot sizing and maintenance strategy for multi-machines system with energy consumption

This paper proposes an integrated model for multi-machines dynamic lot sizing aiming to produce a single item, considering the energy consumption during the production horizon. The objective is to find, firstly, the optimal lot size as well as the number of machines that satisfy a random demand under given service level and secondly, maintenance plan depended to production planning to minimise the total production, energy and maintenance costs. In fact, the problem of energy consumption is one of the most evoked topics especially with the decision of many governments to reduce theirs (For example France is willing to reduce the total consumption by 20% by 2020). The keys of this study are to consider, firstly, the correlation between the forecasting of demand, the variation of the working machines as well as their production rates under energy constraint and secondly the correlation between the production cadences and the maintenance strategy of all machines.     

Setting optimal production lot sizes and planned lead times in a job shop

In this research, we model a job shop that produces a set of discrete parts in a make-to-stock setting. The intent of the research is to develop a planning model to determine the optimal tactical policies that minimise the relevant manufacturing costs subject to workload variability and capacity limits. We consider two tactical decisions, namely the production lot size for each part and the planned lead time for each work station. We model the relevant manufacturing costs, entailing production overtime costs and inventory-related costs, as functions of these tactical decisions. We formulate a non-linear optimisation model and implement it in the Excel spreadsheet. We test the model with actual factory data from our research sponsor. The results are consistent with our intuition and demonstrate the potential value from jointly optimising over these tactical policies.     

On the optimal lot-sizing and scheduling problem in serial-type supply chain system using a time-varying lot-sizing policy

In this paper, we solve the optimal sequencing, lot-sizing and scheduling decisions for several products manufactured through several firms in a serial-type supply chain so as to minimise the sum of setup and inventory holding costs while meeting given demand from customers. We propose a three-phase heuristic to solve this NP-hard problem using a time-varying lot- sizing approach. First, based on the theoretical results, we obtain candidate sets of the production frequencies and cycle time using a junction-point heuristic. Next, we determine the production sequences for each firm using a bin-packing method. Finally, we obtain the production times of the products for each firm in the supply chain system by iteratively solving a set of linear simultaneous equations which were derived from the constraints. Then, we choose the best solution among the candidate solutions. Based on the numerical experiments, we show that the proposed three-phase heuristic efficiently obtains feasible solutions with excellent quality which is much better than the upper-bound solutions from the common cycle approach.     

Single machine multi-product capacitated lot sizing with sequence-dependent setups

In production planning in the glass container industry, machine-dependent setup times and costs are incurred for switch overs from one product to another. The resulting multi-item capacitated lot-sizing problem has sequence-dependent setup times and costs. We present two novel linear mixed-integer programming formulations for this problem, incorporating all the necessary features of setup carryovers. The compact formulation has polynomially many constraints, whereas the stronger formulation uses an exponential number of constraints that can be separated in polynomial time. We also present a five-step heuristic that is effective both in finding a feasible solution (even for tightly capacitated instances) and in producing good solutions to these problems. We report computational experiments.     

Asynchronous teams for joint lot-sizing and scheduling problem in flow shops

The integrated production scheduling and lot-sizing problem in a flow shop environment consists of establishing production lot sizes and allocating machines to process them within a planning horizon in a production line with machines arranged in series. The problem considers that demands must be met without backlogging, the capacity of the machines must be respected, and machine setups are sequence-dependent and preserved between periods of the planning horizon. The objective is to determine a production schedule to minimise the setup, production and inventory costs. A mathematical model from the literature is presented, as well as procedures for obtaining feasible solutions. However, some of the procedures have difficulty in obtaining feasible solutions for large-sized problem instances. In addition, we address the problem using different versions of the Asynchronous Team (A-Team) approach. The procedures were compared with literature heuristics based on Mixed Integer Programming. The proposed A-Team procedures outperformed the literature heuristics, especially for large instances. The developed methodologies and the results obtained are presented.