A hierarchical approach for the capacitated lot-sizing and scheduling problem with a special structure of sequence-dependent setups

This research deals with the single machine multi-product capacitated lot-sizing and scheduling problem (CLSP) with sequence-dependent setup times and setup costs. The CLSP determines the production quantities and the sequence to satisfy deterministic and dynamic demand during multiple periods. The objective is to minimise the total sum of the inventory holding costs and the sequence-dependent setup costs. We consider a special form of sequence-dependent setup times where the larger product we produce next, the more setup time we need. As a solution approach, we propose a two-level hierarchical method consisting of upper-level planning and the lower-level planning. In the upper-level planning, we solve the lot-sizing problem with estimated sequence-independent setup times utilising the characteristic of the special structure of setup times. Then we solve the scheduling problem in the lower-level planning. The proposed method is compared with the single-level optimal CLSP solution and an existing heuristic developed for the uniform structure of setup times.     

Capacitated dynamic lot sizing with capacity acquisition

One of the fundamental problems in operations management is determining the optimal investment in capacity. Capacity investment consumes resources and the decision, once made, is often irreversible. Moreover, the available capacity level affects the action space for production and inventory planning decisions directly. In this article, we address the joint capacitated lot-sizing and capacity-acquisition problems. The firm can produce goods in each of the finite periods into which the production season is partitioned. Fixed as well as variable production costs are incurred for each production batch, along with inventory carrying costs. The production per period is limited by a capacity restriction. The underlying capacity must be purchased up front for the upcoming season and remains constant over the entire season. We assume that the capacity acquisition cost is smooth and convex. For this situation, we develop a model which combines the complexity of time-varying demand and cost functions and of scale economies arising from dynamic lot-sizing costs with the purchase cost of capacity. We propose a heuristic algorithm that runs in polynomial time to determine a good capacity level and corresponding lot-sizing plan simultaneously. Numerical experiments show that our method is a good trade-off between solution quality and running time.     

Comparative analysis of lot-sizing models for multi-stage systems: a simulation study

In the past few years researchers have given considerable attention to various aspects of lot-sizing for single-stage and multi-stage systems in material requirements planning (MRP). Numerous models have been developed and tested on problems with finite horizons and deterministic time-varying demand. A real production system is, however, so complex that no model can capture all the elements under consideration. Instead of the static horizon commonly assumed by researchers, real planning is usually carried out on rolling horizons with different lengths. Such performance measures may include minimization of the total cost, inventory level, and schedule instability caused by rolling horizon and the number of setups. For large and complex product structures, the conventional approach is to apply a single-stage lot-sizing rule once to every stage of the multi-stage system. The Wagner–Whitin (WW) algorithm does provide a solution to this problem, but the length of the necessary calculations precludes its use in practice. As a result, many researchers have proposed numerous other lot-sizing procedures. The purpose of this paper is to examine the performance of various multi-stage lot-sizing procedures under rolling horizon environment using simulation. Cost and schedule instability were used as the performance criteria and product structure, demand variabillity, cost structure and planning horizon were considered as independent variables. It has been shown that Silver–Meal (SM) procedure outperforms other procedures such as WW and incremental approach (ICA) in most of the cases. The performance of ICA and SM are better than their respective counterparts Gaither’s rule (GA) and modified Silver-Meal (MSM). The results also indicate that there are differences in the performance of various lot-sizing procedures when applied to two different product structures.     

Rolling horizon production planning for probabilistic time-varying demands

This paper introduces and tests a production planning procedure to be applied in a rolling horizon with probabilistic demands, based on the work of Bookbinder and Tan (1983). First, the case of no lead time for replenishment is considered and trend-seasonal demands are studied. We vary the set-up cost, order cycle and number of periods in the future for which demand forecasts are available. The procedure is compared to that of Silver (1978) in terms of cost performance, percentage of demand short/period and percentage of periods with stock-outs. Each approach appears to have merit. The proposed procedure generally yields a lower solution cost, while Silver's procedure usually has a lower percentage of demand short/period and a smaller percentage of periods with stockouts. Differences between the two procedures were relatively insensitive to which of three lot-sizing rules was employed within them. Finally, we further extended our procedure to consider non-zero lead times for replenishment and different demand patterns, particularly the normal distribution.     

Statistical aspects of the evaluation of lot-sizing techniques by the use of simulation

An exploratory study was undertaken in order to determine appropriate procedures for statistically analysing output from a stochastic simulation experiment. Single level lot-sizing was studied in an environment with a fixed planning horizon, zero lead time, no forecast errors, and no initial inventories. Of special interest was the specification of factors such as length of the horizon, average time between orders, coefficient of variation related to demand, and number of observations. To achieve increased accuracy, three variance reduction techniques were investigated.     

Lot-sizing rules and freezing the master production schedule in material requirements planning systems under demand uncertainty

Freezing the master production schedule (MPS) is one of the frequently used methods for reducing schedule instability in multilevel material requirements planning (MRP) systems. The selection of the lot-sizing rule is an important decision in MRP systems and has been shown to significantly influence MRP performance. This study examines the interaction between the selection of lot-sizing rules and the selection of MPS freezing parameters under rolling time horizons under different demand uncertainty conditions. The findings of this study enhance the understanding of multilevel MRP performance in a more realistic environment and help MRP practitioners select the proper lot-sizing rule and MPS freezing parameters.     

Decentralised capacitated planning with minimal-information sharing in a 2-echelon supply chain

This paper aims at modelling decentralised planning at the tactical level, with minimal-information sharing coordination, in a 2-echelon supply chain with multiple actors at each echelon. Suppliers manage production and storage at the upstream echelon, while retailers manage transportation and storage at the downstream echelon. The main features of the planning process are (1) decentralisation and coordination using contracts and sharing of only order/supply proposals, and (2) iteration on a rolling horizon. Actor planning is modelled as a capacitated lot-sizing problem on a finite horizon, with the focus on quality of service. The objective is to minimise costs, with a high lost sales penalty if demand is not met. Two other decision problems are pointed out and modelled with Mixed Integer Programming: (1) lost sales allocation between the retailers when their demands cannot be satisfied; and (2) allocation of orders between the suppliers. A multi-agent system combines simulation of the planning process and optimisation of the local decision processes. Several strategies, including retailers’ beliefs about suppliers’ production capacity are proposed and experimentally tested, with two patterns of production capacities. The results compare the proposed allocation strategies and highlight the relevance of the proposed framework for the studied planning problem.     

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.     

Mitigating forecast errors by lot-sizing rules in ERP-controlled manufacturing systems

As enterprise resource planning (ERP) becomes the dominant management software in manufacturing and distribution systems in various industries, some problems associated with its origin, material requirements planning (MRP), still need to be resolved. We examine the effect of forecasting errors, one of the common operational problems in any business operation, in the context of an ERP-controlled manufacturing system. We consider a mitigating remedy, the use of a lot-sizing rule, to cope with the consequences of forecasting inaccuracy without resorting to costly inventory-oriented buffers. An ERP-controlled manufacturing system is simulated to see how these lot-sizing rules mitigate the forecast errors and subsequently generate acceptable system performance. The simulation results should help ease ERP users’ fear of committing another fatal error in demand forecasts, instead encouraging them to consider proper lot-sizing rules to cope with forecast errors.     

Dynamic capacitated lot sizing with random demand and dynamic safety stocks

We present a stochastic version of the single-level, multi-product dynamic lot-sizing problem subject to a capacity constraint. A production schedule has to be determined for random demand so that expected costs are minimized and a constraint based on a new backlog-oriented δ-service-level measure is met. This leads to a non-linear model that is approximated by two different linear models. In the first approximation, a scenario approach based on the random samples is used. In the second approximation model, the expected values of physical inventory and backlog as functions of the cumulated production are approximated by piecewise linear functions. Both models can be solved to determine efficient, robust and stable production schedules in the presence of uncertain and dynamic demand. They lead to dynamic safety stocks that are endogenously coordinated with the production quantities. A numerical analysis based on a set of (artificial) problem instances is used to evaluate the relative performance of the two different approximation approaches. We furthermore show under which conditions precise demand forecasts are particularly useful from a production–scheduling perspective.     

A variable neighbourhood search for integrated production and preventive maintenance planning in multi-state systems

This paper presents a variable neighbourhood search (VNS) to the integrated production and maintenance planning problem in multi-state systems. VNS is one of the most recent meta-heuristics used for problem solving in which a systematic change of neighbourhood within a local search is carried out. In the studied problem, production and maintenance decisions are co-ordinated, so that the total expected cost is minimised. We are given a set of products that must be produced in lots on a multi-state production system during a specified finite planning horizon. Planned preventive maintenance and unplanned corrective maintenance can be performed on each component of the multi-state system. The maintenance policy suggests cyclical preventive replacements of components, and a minimal repair on failed components. The objective is to determine an integrated lot-sizing and preventive maintenance strategy of the system that will minimise the sum of preventive and corrective maintenance costs, setup costs, holding costs, backorder costs and production costs, while satisfying the demand for all products over the entire horizon. We model the production system as a multi-state system with binary-state components. The formulated problem can be solved by comparing the results of several multi-product capacitated lot-sizing problems. The proposed VNS deals with the preventive maintenance selection task. Results on test instances show that the VNS method provides a competitive solution quality at economically computational expense in comparison with genetic algorithms.     

A dynamic lot-sizing model with multi-mode replenishments: polynomial algorithms for special cases with dual and multiple modes

This paper generalizes the classical dynamic lot-sizing model to consider the case where replenishment orders may be delivered by multiple shipment modes. Each mode may have a different lead time and is characterized by a different cost function. The model represents those applications in which products can be purchased through various suppliers or delivered from a single source using various transportation modes with different lead times and costs. The problem is challenging due to the consideration of cargo capacity constraints, i.e., the multiple set-ups cost structure, associated with a replenishment mode. The paper presents several structural optimality properties of the problem and develops efficient algorithms, based on the dynamic programming approach, to find the optimal solution. The special, yet practical, cases of the two-mode replenishment problem analyzed in this paper are analytically tractable, and hence, the respective problems can be solved in polynomial time.     

A heuristic approach for a multi-product capacitated lot-sizing problem with pricing

We address a multi-product capacitated lot-sizing problem with pricing. The objective is to maximise profit. The problem extends the multi-product capacitated lot-sizing problem (CLSP) found in the literature to include price as a decision variable, demand as a function of price, setup time, and more general holding costs. We present a heuristic procedure that can be used to solve large problem instances quickly with good solution quality. The results of computational testing are presented.     

Impact of lot-sizing on the financial performance of dispatching techniques in an MRP-planned fabrication and assembly environment

By simulation analysis of a hypothetical MRP-controlled fabrication and assembly shop, the financial performance of six dispatching techniques are compared under four lot-sizing techniques, two levels of demand and two levels of nervousness. For each lot-sizing technique under low nervousness and low demand the choice of dispatching technique did not substantially affect financial performance. The choice of lot-sizing technique was much more critical. Under increased nervousness, period order quantity and least total cost generated enough setups to deplete excess capacity at the most heavily loaded workcentre. The choice of dispatching technique became critical.     

Inventory control of raw materials under stochastic and seasonal lead times

Most inventory modelling has assumed stochastic demands and constant lead times. However, here we consider a problem for which the opposite situation holds; namely, there is a known constant demand rate, but lead times are random variables. Moreover, the probability distributions of the lead times change in a seasonal fashion. Also, shortages of raw materials result in lost sales. The goal of this paper is to propose heuristic methods for minimizing the expected costs in such a situation. This study was motivated by a problem of management of raw material at a sawmill.     

Purchasing Policy of New Containers Considering the Random Returns of Previously Issued Containers

A number of organizations sell products in containers that can be reused. The time from issue to return of an individual container is usually not known with certainty and there is a chance that the container is never returned (because of loss or irrepairable damage). Consequently, even under a level demand pattern new containers must be acquired from time to time. In this paper a purchasing policy of these new containers is determined for a finite time horizon so as to minimize the total purchasing and expected carrying costs under a prescribed service level. The associated stochastic model is reduced to a deterministic, dynamic lot-sizing problem with possible occurrence of negative net demand (demand minus return). A transformation into the usual nonnegative demand case allows us to apply well-known deterministic lot-sizing procedures to obtain the solution.     

Examining the impact of demand lumpiness on the lot-sizing performance in MRP systems

Material requirements planning (MRP) systems are designed to deal with production scheduling for products with lumpy demand patterns, as opposed to the continuous demand assumed in the classical inventory models. Past MRP lot-sizing studies concluded that the MRP system performance improves as the demand becomes lumpier. The purpose of this paper is to examine the impact of various degrees of demand lumpiness on the system performance of MRP systems by a simulation study. Results show that the performance improves to u certain extent as the demand becomes lumpier. However, the system performance starts to deteriorate when the demand pattern becomes extremely lumpy. MRP users should exercise caution in the introduction of demand lumpiness in the master production schedule (MPS) to induce a better MRP system performance. If a certain degree of demand lumpiness exists in the MPS as given in an operating environment, then the choice of an appropriate lot-sizing rule, such as the Silver-Meal algorithm, could lake advantage of the impact of demand lumpiness.     

Effects of forecast errors on optimal utilisation in aggregate production planning with stochastic customer demand

The hierarchical structure of production planning has the advantage of assigning different decision variables to their respective time horizons and therefore ensures their manageability. However, the restrictive structure of this top-down approach implying that upper level decisions are the constraints for lower level decisions also has its shortcomings. One problem that occurs is that deterministic mixed integer decision problems are often used for long-term planning, but the real production system faces a set of stochastic influences. Therefore, a planned utilisation factor has to be included into this deterministic aggregate planning problem. In practice, this decision is often based on past data and not consciously taken. In this paper, the effect of long-term forecast error on the optimal planned utilisation factor is evaluated for a production system facing stochastic demand and the benefit of exploiting this decision’s potential is discussed. Overall costs including capacity, backorder and inventory costs, are determined with simulation for different multi-stage and multi-item production system structures. The results show that the planned utilisation factor used in the aggregate planning problem has a high influence on optimal costs. Additionally, the negative effect of forecast errors is evaluated and discussed in detail for different production system environments.     

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.