مشکل تعیین اندازه دسته تولید / موجودی تصادفی پویا با محدودیت سطح خدمات

This paper addresses the multi-period single-item inventory lot-sizing problem with stochastic demands under the “static–dynamic uncertainty” strategy of Bookbinder and Tan (Manage. Sci. 34 (1988) 1096). In the static-dynamic uncertainty strategy, the replenishment periods are fixed at the beginning of the planning horizon, but the actual orders are determined only at those replenishment periods and will depend upon the demand that is realised. Their solution heuristic was a two-stage process of firstly fixing the replenishment periods and then secondly determining what adjustments should be made to the planned orders as demand was realised. We present a mixed integer programming formulation that determines both in a single step giving the optimal solution for the “static–dynamic uncertainty” strategy. The total expected inventory holding, ordering and direct item costs during the planning horizon are minimised under the constraint that the probability that the closing inventory in each time period will not be negative is set to at least a certain value. This formulation includes the effect of a unit variable purchase/production cost, which was excluded by the two-stage Bookbinder–Tan heuristic. An evaluation of the accuracy of the heuristic against the optimal solution for the case of a zero unit purchase/production cost is made for a wide variety of demand patterns, coefficients of demand variability and relative holding cost to ordering cost ratios. The practical constraint of non-negative orders and the existence of the unit variable cost mean that the replenishment cycles cannot be treated independently and so the problem cannot be solved as a stochastic form of the Wagner–Whitin problem, applying the shortest route algorithm.

The study of lot-sizing began with Wagner and Whitin (1958), and there is now a sizeable literature in this area extending the basic model to consider capacity constraints, multiple items, multiple stages, etc. However, most previous work on lot-sizing has been directed towards the deterministic case. The reader is directed to De Bodt et al. (1984), Potts and Van Wassenhove (1992), Kuik et al. (1994) and Kimms (1997) for a review of lot-sizing techniques. The practical problem is that in general much, if not all, of the future demands have to be forecast. Point forecasts are typically treated as deterministic demands. However, the existence of forecast errors radically affects the behaviour of the lot-sizing procedures based on assuming the deterministic demand situation. Forecasting errors lead both to stockouts occurring with unsatisfied demands and to larger inventories being carried than planned. The introduction of safety stocks in turn generates even larger inventories and also more orders. It is reported by Davis (1993) that a study at Hewlett-Packard revealed the fact that 60% of the inventory investment in their manufacturing and distribution system is due to demand uncertainty. There has been increasing recognition as illustrated by Wemmerlov (1989) that future lot-sizing studies need to be undertaken on stochastic and dynamic environments that have at least a modicum of resemblance to reality. Inevitably, the forecast errors have to be taken into account in planning the future lot-sizes. Similar concerns have been expressed by Silver: “One should not necessarily use a deterministic lot-sizing rule when significant uncertainty exists. A more appropriate strategy might be some form of probabilistic modelling.” Silver (1978) suggested a heuristic procedure for the stochastic lot-sizing problem assuming that the forecast errors are normally distributed. A similar heuristic, having a different objective function, was presented by Askin (1981). Bookbinder and Tan (1988) proposed another heuristic, under the “static–dynamic uncertainty” strategy. In this strategy, the replenishment periods are fixed at the beginning of the planning horizon and the actual orders at future replenishment periods are determined only at those replenishment periods depending upon the realised demand. The total expected cost is minimised under the minimal service-level constraint. In this paper, we propose a mixed integer programming formulation to solve the stochastic dynamic lot-sizing problem to optimality under the “static–dynamic uncertainty” strategy of Bookbinder and Tan. The optimal solution to the problem is the (s,S) policy with different values for each period in the time varying demand situation. Where the demand level changes slowly, it is usually satisfactory to use a steady-state analysis with constant S and s values updated once per year. However, this is inappropriate if the average demand can change significantly from period to period. This presents a non-stationary problem where the two control parameters change from period to period. The uncertainty in the timing of future replenishments caused by an (s,S) policy may be unattractive from an operational standpoint. Although the inventory problems with stationary demand assumption are well known and extensively studied, very little has appeared on the non-stationary stochastic demand case. Recently, Sox (1997) and Martel et al. (1995) have described static control policies under the non-stationary stochastic demand assumption in a rolling horizon framework. Sox (1997) presents a mixed integer non-linear formulation of the dynamic lot-sizing problem with dynamic costs, and develops a solution algorithm that resembles the Wagner–Whitin algorithm but with some additional feasibility constraints. Martel et al. (1995) transform the multiple item procurement problem into a multi-period static decision problem under risk. Other notable works on non-stationary stochastic demand adopt (s,S) or base-stock policies and are due to Iida (1999), Sobel and Zhang (2001), and Gavirneni and Tayur (2001). In Iida (1999), the periodic review dynamic inventory problem is considered and it is shown that near myopic policies are sufficiently close to optimal decisions for the infinite horizon inventory problem. In Sobel and Zhang (2001), it is assumed that demand arrives simultaneously from a deterministic source and a random source, and proven that a modified (s,S) policy is optimal under general conditions. Gavirneni and Tayur (2001) use the derivative of the cost function that can handle a much wider variety of fluctuations in the problem parameters. The above studies have adopted either static control policies in a rolling horizon framework or dynamic control policies like (s,S), although the static-dynamic uncertainty model is a more accurate representation of industrial practice as pointed out by Sox (1997). Most companies use MRP in some form for their production planning and thus ordering on suppliers. They typically issue advance schedules of requirements. In talking to companies in the supply chain who are in this situation a common complaint is that their customers continually change their schedules, the timing of orders as well as the size of orders. It is the changing of the timing that they find worse. This issue of system nervousness is an active current research area. If there is to be more co-operation and co-ordination in supply chains, then a model that attempts to determine a schedule for the timing of orders in advance taking account of the stochastic demand, which remains fixed, is a contribution of practical interest. This need to fix the deliveries in advance, whilst allowing reasonable flexibility in the order size has been at the heart of the problem of buying raw materials on fluctuating price markets. You have to determine which future months or half months in which to have your delivery of wheat or cocoa, etc. Once this is fixed, suppliers do not allow it to be changed, although the quantities ordered can be varied to some extent. Waiting until the month or half month of delivery and using the standard (s,S) model to decide whether or not to place an order and then its size is not a good option, as the price you pay tends to be much higher as sellers know you are desperate for the material. Thus, investigating models and solution procedures for the static-dynamic uncertainty strategy is potentially important from the practical application perspective.

In this paper, the stochastic dynamic lot-sizing problem with service-level constraints has been modelled under the “static–dynamic uncertainty” strategy of Bookbinder and Tan. A mixed integer programming model for the approach has been formulated. This gives the optimal solution allowing the simultaneous determination of the number and timing of the replenishments and the information necessary to determine the size of the replenishment orders, from the replenishment levels for the periods when stock reviews will take place. Unlike the Bookbinder and Tan model this new MIP model includes a unit variable purchasing/production cost. This model allows an estimation of the accuracy of the Bookbinder and Tan heuristic for solving the “static–dynamic uncertainty” approach to be made, by setting the unit cost equal to zero. If the demand data sets, coefficients of variation and relative holding cost to ordering values used in the numerical experiments could be regarded as typical of what occurs in practice we could conclude that the BT heuristic has a cost performance that is close to the optimal solution. Overall, it gave the optimal solution in 36% of cases and had an average penalty cost of 1.34%. Compared to the optimal solution, which is a mixed integer programming model, the solution times are fast. However, the penalty cost for the BT heuristic will be higher for a non-zero unit purchase/production cost. Moreover, in such cases the problem cannot be modelled as a stochastic form of the Wagner–Whitin problem, treating the replenishment cycles independently and applying the shortest route algorithm. Although, it has been assumed that the replenishment lead-time is zero, it is possible to extend the model for the non-zero replenishment lead-time situation without any loss of generality. A similar model, incorporating the shortage cost, in place of service-level constraints is currently being developed by the authors. Further work could be invested in evaluating the performance of the optimal “static–dynamic uncertainty” strategy in the rolling horizon environment compared to Bookbinder and Tan's heuristic