Anders Thorstenson

An inventory model with Poisson demands and emergency orders

Abstract In this paper we explicitly consider the opportunity to use an emergency supply mode to hedge against demand uncertainty when replenishing a single-item inventory. Normal orders with a relatively long and constant lead time are controlled by a standard ( R , Q ) policy. These orders can only be issued when no other orders are outstanding. When a normal order is outstanding, emergency orders are controlled by a reorder point r(j) and an order-up-to level u(j) , where j is a measure of the time remaining until the normal order is delivered. The emergency orders have a short lead time and they may also have different ordering costs compared to normal orders. Demands not satisfied immediately from inventory are backordered. We formulate a long-run average cost model that includes ordering costs for the two types of orders, backordering costs, and holding costs. Backordering costs are considered both as a unit-based cost and as a cost rate. A tailor-made policy-iteration algorithm is designed and utilized to minimize the inventory cost rate with state-dependent emergency orders. For comparisons, an algorithm for finding a best simple emergency-order policy is also implemented. Simulation is used to check the validity of our model. Numerical results are presented for a set of parameter variations and compared to results without emergency orders as well as to results from an earlier model in the literature. Our results show that substantial cost savings might be obtained by using emergency orders, especially when backordering costs are high. However, the marginal gains are fairly small if state-dependent emergency orders are used instead of a simple emergency-order policy.

A procedure-oriented generic bill of materials

Abstract Bill of Materials (BOM) structures describe the component structure of a product, most often as a tabular structure implemented within a relational database system. The conventional approach is to design a unique BOM for each model or variant of a product. This becomes impossible with a customer-oriented production, where the number of variants may go into high numbers. The solution is to describe and maintain one generic structure that encompasses all variants of a product. This structure may then be used to generate a BOM for each particular variant. Earlier attempts to develop a generic BOM system have been directed towards improving the conventional tabular structures. These efforts have several limitations, mainly because the tabular structure does not offer the flexibility needed to describe generic structures. We propose a generic structure based on a programming language notation. This notation makes it possible to describe the set of possible variants of a product by handling both functional and structural relations between components. The user can explode (execute) any part of the BOM to define a specific product variant. User specifications are given dynamically, as the generic BOM is executed. The system will then automatically generate a specific BOM for this particular product variant. A prototype system has been developed.

A comparison between the order and the volume fill rate for a base-stock inventory control system under a compound renewal demand process

The order fill rate (OFR) is sometimes suggested as an alternative to the volume fill rate (VFR) (most often just denoted fill rate) as a performance measure for inventory control systems. We consider a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered. For this policy, we develop exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process. We also elaborate on when the OFR can be interpreted as the (extended) ready rate. For the case when customer orders are generated by a negative binomial distribution, we show that it is the size of the shape parameter of this distribution that determines the relative magnitude of the two fill rates. In particular, we show that when customer orders are generated by a geometric distribution, the OFR and the VFR are equal.

Optimal (r, Q) inventory policies with Poisson demands and lost sales: discounted and undiscounted cases

Abstract We consider a continuous review (r, Q) inventory system with Poisson demands and at most one order outstanding. The replenishment lead time is either constant or exponentially distributed. Demands not covered immediately from inventory are lost. Costs include a linear order cost with a fixed cost per order, and a fixed cost per unit lost sale. As regards inventory holding costs, the cost of capital often constitutes a major part. This paper focuses on these interest-related holding costs. In the undiscounted case, holding costs are linear and inventory performance is measured by the long-run average total cost incurred per unit time. In the discounted case, the performance measure is the expected present value of the ordering and lost sales costs. The cost associated with capital tied up in inventory is accounted for by an appropriate discount rate. We formulate an exact model and design a policy-iteration algorithm for the discounted case. Results on the form of an optimal replenishment policy are derived and the model is compared to a previously derived model for the undiscounted case. Numerical experiments are used to evaluate the difference between the optimal solutions with and without discounting. The effect of a stochastic lead time on this difference is also considered by comparing solutions with constant and exponential lead times. In general, the differences seem to be fairly small but exceptional cases exist when the service level is low.

Optimal and approximate (Q, r) inventory policies with lost sales and gamma-distributed lead time

Abstract We consider the continuous review inventory control system with fixed reorder point r and constant order quantity Q . Demands are assumed to be generated by a Poisson process with one unit demanded at a time. Demands not covered immediately from inventory are lost. For the case of at most one order outstanding we derive and implement a model to obtain exact solutions for the reorder point and the order quantity. The model is formulated as a semi-Markov decision model and we show that if it is profitable to issue orders then a ( Q , r ) policy is average-cost optimal. In general neither a ( Q , r ) policy nor an ( policy is optimal if demand for more than one unit at a time is allowed in our model. A policy-iteration algorithm is developed for finding the optimal policy. We focus on the shape of the lead-time distribution by studying the optimal policy when the lead times are gamma distributed with different shape parameters. The results are compared to those obtained when applying approximate methods to the reorder-point inventory system.

Solving a combined cutting-stock and lot-sizing problem with a column generating procedure

In NonAs and Thorstenson [A combined cutting stock and lot sizing problem. European Journal of Operational Research 120(2) (2000) 327-42] a combined cutting-stock and lot-sizing problem is outlined under static and deterministic conditions. In this paper we suggest a new column generating solution procedure for this problem that works well on both small and large-sized problems. The procedure includes characteristics from both the column generating procedure in NonAs and Thorstenson, which works well on small-sized problems, and from the sequential heuristic due to Haessler [A heuristic programming solution to a nonlinear cutting stock problem, Management Science 17(12) (1971) 793-802], which works well on large-sized problems. Numerical results are presented that show that the new heuristic performs better than both of the earlier procedures. Comparisons with results obtained by other authors indicate that the procedure works well also for the extended cutting-stock problem with only a setup cost for each pattern change.

Comparing sourcing strategies in two-echelon supply chains

We investigate four sourcing models with respect to either cooperative or non-cooperative planning strategies and either sole or dual sourcing. A two-stage supply chain is considered. It involves a single buyer and either one or two supplier(s)/vendor(s). At the buyer, the product is consumed at a constant rate and an (r, Q) inventory control policy is used for replenishments. The delivery lead time from the vendors is stochastic. The cost function comprises five elements: inventory holding costs for buyer and vendors, backorder costs and ordering costs for the buyer, and setup costs for the vendors. The objective is to minimize total system costs incurred at the buyer and the vendors. As there is no overall dominating combined sourcing strategy, a major finding is that determining the best strategy requires a detailed analysis. However, when total system costs are taken into account, dual sourcing does not appear as beneficial as sometimes claimed in the literature on order splitting.

A multi-phase algorithm for a joint lot-sizing and pricing problem with stochastic demands

Stochastic lot-sizing problems have been addressed quite extensively, but relatively few studies also consider marketing factors, such as pricing. In this paper, we address a joint stochastic lot-sizing and pricing problem with capacity constraints and backlogging for a firm that produces a single item over a finite multi-period planning horizon. Thece-dependent demands. The stochastic demand is captured by the scenario analysis approach, and this leads to a multiple-stage stochastic programming problem. Given the complexity of the stochastic programming problem, it is hard to determine optimal prices and lot sizes simultaneously. Therefore, we decompose the joint lot-sizing and pricing problem with stochastic demands and capacity constraints into a multi-phase decision process. In each phase, we solve the associated sub-problem to optimality. The decomposed decision process corresponds to a practically viable approach to decision-making. In addition to incorporating market uncertainty and pricing decisions in the traditional production and inventory planning process, our approach also accommodates the complexity of time-varying cost and capacity constraints. Finally, our numerical results show that the multi-phase heuristic algorithm solves the example problems effectively.

Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times

We show that well-known textbook formulae for determining the optimal base stock of the inventory system with continuous review and constant lead time can easily be extended to the case with periodic review and stochastic, sequential lead times. The provided performance measures and conditions for optimality are exact

Product variants and the turnover rate

The objective of this paper is to investigate the impact of the number of product variants on the turnover rate in a production-inventory system. The original problem arose in a Swedish firm which manufactures candy. The model set forth includes a two-stage production system consisting of stocks for raw material, two consecutive manufacturing facilities with in-between buffer stocks, and final inventories for finished goods. The way in which the turnover rate is affected by the number of end-product variants is discussed and the results derived theoretically are compared with empirical data