بررسی های دوره ای مشکل اندازه زیاد با بازده تصادفی، اختلالات و ظرفیت موجودی

This paper examines a periodic-reviewed lot sizing problem with random yields, disruptions and limited inventory capacity. To characterise the continuous production, an additive random yield model is considered rather than a multiplicative one. Disruptions cause breakdowns to production. Inventory capacity is included since the production has to be shut down when the inventory buffer is full. Both disruptions and shutdowns lead to a start-up cost and a stochastic lead time to recover the production. These compound factors of uncertainty are encountered in practical planning decisions in process industries. We review the existing random yield models, which are then compared with the additive model. With a linear production cost, the additive model has an order-up-to policy to be optimal. Disruptions deteriorate the expected actual production quantity and the fill-rate dramatically, even though the optimal order-up-to level increases compared with the cases of no disruption. Considering inventory capacity makes the problem to be a non-convex dynamic programming problem. Numerical analysis shows that the performance is dramatically deteriorated when the inventory capacity is rather tight, which indicates the importance of selecting a proper inventory capacity to reduce the negative impacts and avoid redundant investment on capacity. Moreover, the start-up cost plays an important role in determining the level of inventory capacity.

Process industry generally produces in a continuous manner. Different from a discrete production, a continuous production often needs more time and spends more resources for starting up the production, therefore its production should ideally remain in a running state for a very long period of time before the complete overhaul of the system. However, the production is also affected by various uncertainties. First, the production quantity cannot be precisely controlled as in a discrete production process. The actual output often deviates from the target production quantity due to various factors (in a chemical process, for instance the ambient temperature, source of raw materials). Second, the production could be interrupted due to the unexpected disruptions of local or preceding production facilities or utility disturbances (Lindholm and Johnsson, 2012). Third, a continuous production system often consists of several production processes (facilities), which are often linked with buffer tanks. As intermediate products are stored in these tanks, there is a physical restriction on the inventory volume. Once a tank is full, the preceding production is forced to be shut down. All these factors also interact with each other, and contribute to the compound random yields of production. Hence, in a process industry production planning should take into account the factors mentioned above in order to balance demand and supply between the processes and maintain a smooth flow of materials. We observe the above planning challenges from the practice of a chemical products group in our research project of Swedish process industries. This group owns three plants. Each plant consists of several production areas. A target production quantity is determined for each production area every day. The output products are liquid and stored in a large buffer tank after production. Demand is usually known for the future 10–14 days and met by the yields and inventory in the tank. Unsatisfied demand is backlogged. As lack of advanced planning tools, the current planning method is mainly to use the average demand of the planning horizon (10–14 days) as the production quantity. Due to the randomness in production as mentioned before, manual adjustments based on personal experience are often made. The relation between production quantity and system performance in terms of cost and fill-rate is not clear. The operations efficiency has a great potential to be improved. In this paper, we study a multi-period lot sizing problem with random yields, disruptions and a restriction of the inventory capacity. There are three critical questions to be answered: First, how these factors affect the production decisions? Second, how these factors should be incorporated into the production planning decision? Third, in which circumstances the system performance will be significantly improved by considering these factors which also complicate the decision process? This planning problem is actually a periodic lot-sizing problem with random yields. Since lot-sizing problem was first studied by Harris (1913), more practical and complex issues have been considered. Yano and Lee (1995) give an extensive review on the lot-sizing problem with random yields. One main way is to model the random yield to be stochastically proportional to the target quantity, and this leads to a multiplicative form of yield model. It is actually originated from the concern of producing defective items (Henig and Gerchak, 1990). With such a multiplicative model, the variance of random yields increases as the target quantity. In the above mentioned process industry, the main uncertainties come from utility disturbances and turbulences of production processes which are mostly independent of the target quantity. This argument is also supported from the analysis of the target and the real output data collected in the case company. In this paper, we will focus our study on an additive random yield model. Random production capacity, as described in literature, is another reason for producing a random output of a production system. The actual yield in this case can be modelled as the minimum of the target quantity and the realisation of random capacity. In a process industry, the utility and buffer tank can be considered as the production resources. Utility disruptions cause breakdowns of production, which leads to loss of production capacity. Hence, the unstable utilities can be viewed as random production capacity. The physical limitation of storage space constraints the inventory as well as the certain production quantity. By using the additive random yield model, we capture the feature that the yield could be greater than the target production quantity, and consequently the buffer tank could be potentially full and the preceding production will be shut down. Moreover, recovering from disruptions or shutdowns induces a large start-up cost and a random recovery lead time, which are also included in this paper. The remainder of the paper is organised as follows. The related random yield models are reviewed and compared with an additive model in Section 2. Then we include disruptions in Section 3 and inventory capacity in Section 4. The conclusions are drawn in Section 5.

The paper examines a multi-period lot sizing problem. Inspired by the practice in a process industry, we incorporate the random yields, disruptions and limited inventory capacity into the planning decisions. To highlight the characteristics of continuous production, we employ a model with an additive random yield rather than a multiplicative one in literature, and also include an inventory capacity rather than a production capacity. Both disruptions and full inventory capacity will stop production, which incurs a huge start-up cost and a stochastic recovery lead time. These practical features have not been examined jointly in the production planning. We first review the existing random yield models. We then analyse the additive random yield model and compare it with the multiplicative models in literature. Under general conditions, all the models have a critical level policy to be optimal. Furthermore, if the production cost is linear, the additive model has a myopic order-up-to policy to be optimal, which facilitates the implementation. With disruptions, the optimal order-up-to level is decreasing along with the time during the planning horizon. With inventory capacity, the order-up-to policy is still optimal, while the problem becomes a non-convex dynamic programming problem so that there is no explicit expression for the optimal order-up-to level. The theoretical part is summarised in Table 5. Table 5. Summary of optimal policies in this paper. Factors Optimal policy Order-up-to level Additive random yield Order-up-to level policy, if production cost is linear Myopic Disruptions Monotone decreasing Inventory capacity N/A Table options Two factors, disruptions and inventory capacity, are further investigated. Due to disruptions, the actual output quantity is low, resulting in low fill-rate. The start-up cost of disruptions has no impact on performance but a long recovery lead time can deteriorate the performance dramatically. With inventory capacity, the optimal policy is still order-up-to while the optimal order-up-to level cannot be obtained easily due to the non-convex problem. By numerical experiments, we observe that the impact of inventory capacity on performance becomes significant and negative dramatically when the capacity becomes relatively tight. The start-up cost in this case also has obvious impacts on performance. As in practical planning, these two factors are often ignored. We imitate the practical planning by developing hybrid policies which make decisions without considering these factors but implement the solutions in the environment with disruptions and inventory capacity. The results show that the hybrid policy only generates near-optimal solutions when randomness is negligible. Otherwise, it is valuable to use complex models to obtain the optimal policy. Further extensions can be made by studying multi-echelon systems where multiple production areas are connected by buffer tanks with limited capacity. In this case, the empty tank can force the successive production to shut down. Understanding the structure of optimal production policies would provide useful insights to maintain the operations of an efficient and robust production system.