تجزیه و تحلیل مدل های موجودی سفارش دسته ای همراه با هزینه راه اندازی و محدودیت ظرفیت

Stochastic periodic-review batch ordering inventory problems appear in many industrial settings. However, few literature deals with the optimal ordering polices for such problems, no mention to the inclusion of the fixed ordering cost and the production capacity. In this paper, we consider a single-item periodic-review batch ordering inventory system with the consideration of the setup cost and the capacity constraint for each order over a finite planning horizon. By proposing several new convex notions, we show that a batch-based (s,S) policy is optimal for the unlimited ordering capacity case, while for the limited ordering capacity case, a modified (r,Q) policy is optimal for the setting with zero ordering setup cost, and a batch-based X–Y band policy for the setting with positive ordering setup cost. Moreover, we analytically study the sensitivity of the policy parameters with respect to the capacity and batch order size, and derive the bounds on the optimal policy parameters. We further extend our analysis to the infinite horizon setting and show that the structure of the optimal policy remains similar. Finally, the numerical experiments provide some insights into the impact of model parameters on the benefit of reducing the batch size and increasing the ordering capacity, and indicate that ignoring batch requirement may lead to a significant cost increment.

Traditional research on inventory management mainly focuses on studying ordering policies for continuous order size, i.e., the ordering quantity is infinitely dividable. However, in practical operations, e.g., materials usually flow at fixed batch sizes. For example, consumer packaged goods typically arrive at retailing stores in casepacks (Ketzenberg et al., 2000), finished goods may be transported in full containers from manufacturers to distributors, and work-in-process (WIP) is often processed in some convenient lot sizes between production stages. Despite that, inventory models with batch ordering models are still relatively understudied, as those models cannot produce nicely structured policies as compared to their continuous counterparties. Moreover, the complexities in practice go far beyond the ordering batch size. Two other common factors, i.e., setup cost and capacity constraint, further complicate the ordering decisions in the production–inventory systems. The ordering setup cost is also known as the fixed cost as opposed to the variable cost. Taking the retailing store for example, the setup cost may include the search cost for a counter party, the cost for paperwork, the cost of transportation, etc. The ordering capacity is due to shortage of capital or limited resource of the retailer, or the production capacity of the supplier. In practice, the ordering capacity always stays stable or changes only slightly over time, as it normally takes a significant amount of time for suppliers to adjust his capacity. Thus, the ordering capacity can be (or approximately) viewed as a constant and observable. Although inventory models dealing with the two issues (i.e., setup cost and capacity constraint) independently are extensive, those considering these two issues together are scarce, not to mention inventory models considering setup cost, capacity constraint and batch order size at the same time. This paper seeks to fill the gap in the literature by studying the optimal policies for a capacitated inventory system with batch ordering and setup cost. Specifically, in this paper, we consider a single-item periodic-review capacitated inventory system with batch ordering and setup cost over a finite planning horizon. At the beginning of each period, the firm first observes his initial inventory level, and then decides the ordering quantity, which should be in batch size and is constrained by the ordering capacity, to raise the inventory level so as to satisfy the demand at the end of this period. Each order will occur at a setup cost with its leadtime equal to zero. After demand realizes, the excess inventory will be taken over to the next period, while the unsatisfied demand will be backlogged. We derive the optimal ordering policies to minimize the total expected cost over the whole planning horizon. By proposing several new convex notions, we show that a batch-based (s ,S ) policy, i.e., (s,S)Q(s,S)Q policy, is optimal when the ordering capacity is unlimited, while when the ordering capacity is limited, a modified (r ,Q ) policy is optimal for the case with zero setup cost and a batch-based X –Y band policy, i.e., [X−Y]Q[X−Y]Q policy, is optimal for the case with positive setup cost. Moreover, we analytically investigate the sensitivity of the policy parameters with respect to the capacity and batch order size, derive the bounds on the optimal policy parameters, and numerically study the value of reducing batch size and expending system capacity, investigate the loss by ignoring batch size to constitute the managerial contribution of this paper.

In this paper, we consider a single-item periodic-review capacitated inventory system with batch ordering and setup cost over a finite planning horizon. We characterize the optimal ordering policies for the firm whose objective is to minimize his total expected cost over the planning horizon. With the assistance of some new convex notions, namely Q-jump-K-convexity and Q-jump-(C,K)-convexity, we find that a batch-based (s,S) policy is optimal when the ordering capacity is unlimited, while when the ordering capacity is limited, a modified (r,Q) policy is optimal for zero setup cost and a batch-based X–Y band policy is optimal for positive setup cost. Moreover, we analytically study the sensitivity of the policy parameters with respect to the capacity and batch size, and derive the bounds on the optimal policy parameters. We also conduct some numerical experiments to provide some insights into the impact of model parameters on the benefit of reducing the batch size and indicate that ignoring batch requirement may lead to a significant cost increment. For future research, an important challenge is to take the price decision into account. In this case, the analysis would become excessively complicated, even for the model without batch ordering. Furthermore, the reality is more complex than what this paper considers. For example, in some industries, the capacity is variable over the planning horizon; the ordering cost is also random; there are multiple ordering sources, each of which has different batch size, setup cost and variable cost, etc. Thus, another direction of future research is to further derive the optimal ordering policies by including these reality factors into consideration.