خصوصیات قوانین تعیین اندازه دسته تولید تحت تقاضای ناهنجار

When a lot-sizing problem is viewed from a context of buyer–supplier relationships, an important phenomenon frequently encountered by the supplier is a so-called lumpy demand. However, there has been very little attention devoted to study the behaviour and performances of lot-sizing rules under the situation of lumpy demand. This paper presents both analytical and experimental studies of lot-sizing rules for lumpy demand situations. The analytical study is based on the assumption that constant demand occurs for every fixed number of periods. In the experimental study, both the quantity of and the time between demands are allowed to vary. The studies show that analytical results provide good insights in understanding the behaviour and performances of lot-sizing rules when more realistic situations are addressed in the experimental study. The paper also confirms that the results of lot-sizing studies under the situation of non-lumpy demand cannot be entirely generalised to the situation with lumpy demand.

In a supply chain the first-level supplier/producer receives the stochastic demands of the consumers on a continuous time basis. If the level of demand is not sufficiently stable over time so a JIT production system is appropriate, the supplier will have to produce in discontinuous lots. This means that the second tier suppliers, who provide the materials and components needed by the first level supplier, will have to deal with the phenomenon of so-called lumpy demand. The term lumpy demand is used to represent the situation where a demand for an item does not occur every period, but rather, there is a large proportion of periods having zero demand. Lumpy demand received by a supplier will arise as a result of a buyer using an inventory ordering system or a deterministic lot-sizing procedure in an ERP or MRP system which lead to a decision not to order every period. The payment terms offered by the supplier to its buyers also play a role. As Kingsman 1983 and Kingsman 1991, Carlson and Rousseau (1989), Carlson et al. (1996), and Pujawan and Kingsman (1999) have shown, date terms payment at a fixed time in the month after delivery gives larger and less frequent orders than day terms, where payment is required a fixed time after delivery. Most of the large amount of work on lot-sizing theories over recent decades appears to overlooked the fact that lumpy demand is an important issue in practice and that the behaviour of lot-sizing techniques under non-lumpy demand is not necessarily the same as that under lumpy demand. Lot-sizing rules that provide good solutions under non-lumpy demand might turn out to be unsatisfactory when they are applied in a lumpy demand situation. Experiments described by Blackburn and Millen (1985) showed that the Silver Meal lot-sizing rule, which usually performs well under non-lumpy demand turns out to be less satisfactory when demand becomes lumpier. Some methodological issues on demand lumpiness studies are also still inconclusive. Following the work of Kainmann (1969), many studies have based their measure of the demand profile on the coefficient of variation (CV) of demand per period. Bobko and Whybark (1985) even concluded that CV is a robust measure of demand profile. Although a little modification was introduced in generating the demand pattern, Ho 1993 and Ho 1995 still relied on the coefficient of variation of demand to provide the measure of demand lumpiness. On the other hand, Williams and Peters (1987) commented that the use of CV might not be a sufficient descriptor of the demand profile. Some studies on demand lumpiness have concentrated on the issues of reorder levels (e.g., Williams, 1982; Mak and Hung, 1986) and forecasting (e.g., Croston, 1972; Bartezzaghi et al., 1999). Very few papers have been devoted to study the behaviour of lot-sizing rules such as Silver Meal, Least Unit Cost, Part Period Balancing, etc., under the situation of lumpy demand. Our observation to date reveals that the rare studies on this issue have been conducted by using simulation, for example, Ho 1993 and Ho 1995. However, to obtain a better understanding on the behaviour of the system, an analytical study for some restricted assumptions is often helpful. There appears to be no paper clearly addressing the analytical properties of lot-sizing rules under the situation of lumpy demand. Given the emphasis now on co-ordination and management of the supply chain, the neglect of the effect of lumpiness in demand needs remedying. This paper describes a small part of a major research project in this area by the authors, see Pujawan (2000). Following the approach first introduced by Croston (1972) we distinguish demand profiles on the basis of two separate characteristics, demand lumpiness and demand variability. Demand lumpiness relates to the average inter-arrival time between demands. A set of demands is considered to be lumpy if they do not occur every period. Demand variability refers to the degree of variation of the size of the demands that occur. When demand is lumpy but all positive demands occur at the same quantity, demand variability is zero. The CV of the positive demands will be used to measure demand variability. In this paper, both analytical and experimental studies of lot-sizing rules under the situation of lumpy demand will be presented. Section 2 of the paper presents some analytical models on the behaviour of lot-sizing rules under the situation of simple lumpy demand. Section 3 discusses the results of simulation studies that allow both the inter-arrival times between demands and the demand sizes to be stochastically variable. The final section concludes the paper with some future issues.

Analytical and experimental studies on lot-sizing rules under the situation of lumpy demand have been presented in this paper. Some of the possible implications of the analyses are clearer if one presents the results graphically. Fig. 1 shows the number of orders, average inventory and relative unit costs averaged over all five TBOs for the different lot-sizing rules by demand lumpiness and demand variability. The ICR is excluded from the graph because of its generally poor performance and to show the differences in the inventory levels of the other four rules more clearly. The graphs of unit cost also suggest that the relative efficiency of lot-sizing rules under the situation of lumpy demand differ more significantly compared to those of non-lumpy demand. Hence, the results confirm that the behaviour and the performance of lot-sizing rules under the situation of lumpy demand can not be entirely generalised from the analyses based on a non-lumpy demand situation. The study reveals that analytical properties of lot-sizing rules derived from the assumption of constant demand and time between demand can be used as a good basis in understanding the behaviour of lot-sizing rules under a more realistic situation where both demand and time between demand are not constant. Some modification may be required in ordering for a certain rule to behave efficiently in both situations. It is shown that different ways in calculating the cost per period in the Silver Meal rule could end up with substantially different ordering decisions and cost performances when demand is lumpy. The results of the study endorse strongly the need to modify the Silver-Meal heuristic as here in rule SM2. This confirms the results found by Blackburn and Millen (1985) and shows how the rule may be modified to overcome the problems. The different conclusion to that of Blackburn and Millen on the ICR rule, that it is not to be recommended, arises because the evaluations in this paper covered longer TBOs. The performance measures used are basically the same, but the earlier work used short TBOs. The results of the simulation studies are consistent with Proposition 1a at the end of Section 2 of the analytical analysis, that the ICR gives higher order quantities then the other rules when the orders cover three or more periods with non-zero demand. Although it perhaps needs further confirmation from a greater variety of experiments, the results of the simulations here indicate that it is the demand lumpiness, the relative number of positive to zero demand periods, rather than the demand variability, given by the CV, that affects the relative performance of the different rules on the usual measures of performances. The general shape of the graphs in terms of how demand lumpiness affects the different rules is virtually the same for both low and high CVs of demand. There are large differences between the different demand lumpinesses for the same CV. This suggests that insights gained from the analytical studies on lumpiness, ignoring demand variability, are potentially useful. It is possible that these results may not apply to situations with higher coefficients of variation for the demand size, particularly if the distribution of demand size in “non-normal”. While providing important insights toward the analysis of lot-sizing problems in buyer–supplier relationships, the study needs further developments. An important extension is to investigate a two-level or three-level problem as a whole when both the first-level and second-level suppliers apply some lot-sizing approach, including the use of stochastic inventory models such as (Q,r) or (T,R) systems. When the demand encountered by the first tier supplier (the buyer) from its customers is stochastically uncertain and the buyer applies traditional lot-sizing rules such as the ones used in this paper, the order patterns sent by the buyer to the second tier supplier (the supplier) will be a lumpy demand pattern. It will also be a dynamic situation with the buyer updating its forecasts and changing the planned timing and size of orders. So the buyer's situation will be characterised by uncertainties in both the order quantity and the time between orders. Analysing the effects of different amounts of co-operation and exchange of information is an important next stage in both understanding supply chain behaviour and managing it better for both parties.