Søren Glud Johansen

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.

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.

Optimal and near-optimal policies for lost sales inventory models with at most one replenishment order outstanding

In this paper we use policy-iteration to explore the behaviour of optimal control policies for lost sales inventory models with the constraint that not more than one replenishment order may be outstanding at any time. Continuous and periodic review, fixed and variable lead times, replenishment order sizes which are constrained to be an integral multiple of some fixed unit of transfer and service level constraint models are all considered. Demand is discrete and, for continuous review, assumed to derive from a compound Poisson process. It is demonstrated that, in general, neither the best (s, S) nor the best (r, Q) policy is optimal but that the best policy from within those classes will have a cost which is generally close to that of the optimal policy obtained by policy iteration. Finally, near-optimal computationally-efficient control procedures for finding (s, S) and (r, Q) policies are proposed and their performance illustrated.

Pure and modified base-stock policies for the lost sales inventory system with negligible set-up costs and constant lead times

Abstract The studied inventory system with continuous review has an easily computed optimal (S−1,S) policy when unsatisfied demands are backlogged. We assume that unsatisfied demands are lost and then it is also easy to compute the best (S−1,S) policy. But, as demonstrated by Roger Hill at the ISIR Symposium in 1996, this pure base-stock policy can never be optimal if S⩾2. Our focus is on periodic review. We use Erlangs loss formula to derive approximate expressions for the stockout probability and the average cost. These expressions are used to approximate the average cost and to compute a good base-stock. We formulate and implement a Markov decision model to find the optimal replenishment policy. The model is solved by a policy-iteration algorithm. Because the optimal policy is often rather complicated, we introduce modified base-stock policies. They are specified by a pair (S,t) where S is the base-stock and t is a lower bound for the number of review periods between review epochs in which placing a replenishment order is permitted. A simple one has S equal to the base-stock computed from Erlangs formula and fixes t as the largest integer which is less than or equal to the ratio of the number of review periods per delivery period and S. Our numerical examples show that the simple modified base-stock policy provides most of the cost reduction which can be obtained by replacing the best pure base-stock policy by the optimal policy.

The (r,Q) control of a periodic-review inventory system with continuous demand and lost sales

Abstract In this paper we consider a periodic review inventory model with lost sales during a stockout and with the constraint that at most one replenishment order may be outstanding at any time. Demands in successive review periods are independent, identically distributed variables from a continuous distribution. The fixed lead time is an integral number of review periods. We explore control policies of the (r,Q) type – that is a replenishment order of size Q is placed when the inventory position (stock in hand plus stock on order) falls to or below the re-order level r. We use asymptotic renewal theory results to estimate the `undershoot’ of the re-order level r and also to estimate the cycle stockholding cost (which turns out to take a relatively simple form). Based on these approximations we set out a policy improvement solution methodology and illustrate this with some numerical examples for which demand is normally distributed. These numerical examples suggest that a relatively simple approach, based on the economic order quantity, can provide results which are very close to optimal.

Can-order policies for coordinated inventory replenishment with Erlang distributed times between ordering

Abstract We consider the replenishment of items from the same supplier when the cost of each replenishment order is the sum of a major fixed cost and ordinary item-dependent costs. The items are coordinated by the ( s , c , S ) policy. A decomposition algorithm is developed to compute the parameters of the policy for each item. The algorithm assumes that the time between ordering is Erlang distributed, with parameters depending on the actual policy. The algorithm is used on various examples and it performs well in comparison with other algorithms. We illustrate that this and related decomposition algorithms do not always converge and the policy identified by the algorithms is not necessarily the best one among the ( s , c , S ) policies.

Can-order policy for the periodic-review joint replenishment problem

In this paper we study the stochastic joint replenishment problem. We compare the class of periodic replenishment policies and the class of can-order policies for this problem. We present a method, based on Markov decision theory, to calculate near-optimal can-order policies for a periodic-review inventory system. Our numerical study shows that the can-order policy behaves as well as, if not better than, the periodic replenishment policies. In particular, for examples where the demand is irregular, we find cost differences up to 15% in favour of the can-order policy.

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.

Lot sizing for varying degrees of demand uncertainty

Abstract The average costs for some well-known lot-sizing techniques are investigated by simulation. We study an inventory system with periodic review, instantaneous (or fast) delivery of every replenishment order and backlogging of unfilled demand. At review point t, the amount to be delivered in period t+j consists of firm demand from customers who have ordered j periods or more before their desired delivery in period t+j and normally distributed random demand from customers who order less than j periods in advance. If j is big then all demand is random. We use the profile of advance ordering to specify various degrees of demand uncertainty. If the demand uncertainty is low then the best lot sizes are those computed with a rolling horizon by Deterministic Dynamic Programming. For increased demand uncertainty, an (s, S) policy becomes a better choice. We suggest a policy-iteration algorithm to compute the reorder point sSDP1 and the order-up-to level SSDP1 for the model which assumes that the actual demand is known one period in advance only. We recommend to implement the reorder point sSDP1 and to specify the actual order-up-to level as SSDP1 plus a term adjusting for the neglected actual information about advance orders.