Rationing policies for some inventory systems

This paper considers inventory systems which maintain stocks to meet various demand classes with different priorities. We use the concept of a support level control policy. That is rationing is accomplished by maintaining a support level, say K, such that when on hand stock reaches K, all low priority demands are backordered. We develop four analytical and simulation models to improve the existing models. Firstly, multiple support levels are used instead of using a single support level. Secondly, a simulation model with a more realistic assumption on the demand process has been provided. Thirdly, a single period deterministic cost minimisation model has been developed analytically. Finally, we address a continuous review (Q, r) model with a compound Poisson process.     

An Optimal Heuristic For Coordinated Multi-Item Inventory Replenishments

The inventory control problem can be vastly simplified if the replenishments of inventory items are coordinated with one another. That is, whenever an item is replenished, n other items, where n is a decision variable, are also replenished. One way to ensure this would be to classify the inventory items into several groups with a common order interval for each group. In this paper we establish that the optimal groups will be consecutive by hD/A, where h, D and A are the holding cost, demand rate and set-up cost of an item respectively. Using this property of consecutiveness, we develop a fast converging heuristic to create m groups optimally, m = 2, 3,..., M. The heuristic is a substitute for the dynamic programme which would otherwise be necessary and it has the potential for nomographic applications.     

A Multiple Reorder Point Inventory Policy

This paper discusses the development and application of a multiple reorder inventory policy which can be stated as follows: reorder an optimal lot size Q when inventory (stock on hand) falls to R, R-Q, R-2Q,..., R-NQ; where R is the reorder level. If demands cause the inventory to fall below two reorder levels, say a jump from R+ ? to R-2Q+?′ where ? and ?′ < Q, an order for 2Q is placed. The policy is a form of (S,q) policy where the maximum stock level S = R + Q. The system is of particular value in cases where the coefficient of variation of lead time demand μ _l (μ _l = σ _l /λ _l )is large (say >0·5) and continuous inventory records are maintained. Tables, charts and nomographs to simplify clerical tasks can be obtained quite readily. In this formulation R and Q are not independent factors as in the usual Wilson formulation, but are obtained by minimizing a single cost functional subject to the constraint of a specified risk of out-of-stock condition or a specified level of service (Galliher and Simmond, 1957), (Morse et al., 1959). The particular application concerns the raw material inventories of a manufacturer of metal pressings who is required to offer “immediate service”. The demand distribution during the lead time closely approximates the exponential distribution, and lead times are constant for each raw material. The application of the multiple reorder policy results in a 30 to 35 per cent reduction in inventory for a 95 per cent service level. Measures of sensitivity and response are obtained, and the mean number of shortages is expressed in closed form. The policy is compared with the Wilson policy and shown to be more “effective” in that it results in lower inventories and a smaller number of orders for the case considered.     

Control Theory in Production/Inventory Systems: A Case Study in a Food Processing Organization

Production/inventory systems must attempt to cope with varying demand. This is traditionally done by applying the R,t inventory system after forecasting demand. If the demand is not stationary in mean and variance, this system relies on advanced forecasting models and may still be inadequate. A technique is presented which applies feedback control theory to the maintenance of a chosen level of safety stock. A term involving the rate of change of stock on hand is introduced to overcome the deficiencies of the standard R,t reorder policy when dealing with erratic demand. Once the parameters for an individual application are chosen the technique may be automated on any spreadsheet or modelling package. Results are produced which indicate that this is a superior heuristic to the standard R,t system.     

(<Emphasis Type="Italic">Q,r</Emphasis>) Inventory Model with a Mixture of Lost Sales and Time-Weighted Backorders

This paper presents a stochastic inventory model for situations in which, during a stockout period, a fraction β of the demand is backordered and the remaining fraction 1 – β is lost. The model is suggested by the customers' different reactions to a stockout condition: during the stockout period, some patient customers wait until their demand is satisfied, while other impatient or urgent customers cannot wait and have to fill their demand from another source. The cost of a backorder is assumed to be proportional to the length of time for which the backorder exists, and a fixed penalty cost is incurred per unit of lost demand. Based on a heuristic treatment of a lot-size reorder-point policy, a mathematical model representing the average annual cost of operating the inventory system is developed. The optimal operating policy variables minimizing the average annual cost can be calculated iteratively. At the extremes β = 1 and β = 0, the model presented reduces to the usual backorders and lost sales case, respectively.     

Effects of Centralization on Expected Costs in a Multi-location Newsboy Problem

This is a single-period, single-product inventory model with several individual sources of demand. It is a multi-location problem with an opportunity for centralization. The holding and penalty cost functions at each location are assumed to be identical. Two types of inventory system are considered in this paper: the decentralized system and the centralized system. The decentralized system is a system in which a separate inventory is kept to satisfy the demand at each source of demand. The centralized system is a system in which all demands are satisfied from one central warehouse. This paper demonstrates that, for any probability distribution of a location's demands, the following properties are always true: given that the holding and penalty cost functions are identical at all locations, (1) if the holding and penalty cost functions are concave functions, then the expected holding and penalty costs in a decentralized system exceed those in a centralized system, except that (2) if the holding and penalty cost functions are linear functions, and for any i≠j, P_ij, the coefficient of correlation between the ith location's demand and the jth location's demand is equal to 1, then the expected holding and penalty costs in a decentralized system are equal to those in a centralized system.     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.     

A Graphical Aid for the Initial Purchase of ‘Insurance Type’ Spares

Expensive renewable spares known as ‘insurance type’ spares are often a major concern in the design and setting up of industrial, commercial and military systems. These spares, though low in demand, are critical to the system's operation and their unavailability can lead to excessive downtime costs. Due to their nature, the (S-1, S) inventory control model provides an appropriate replenishment policy for this class of items, where S is the maximum number of spares in inventory. A (S-1, S) model with Exponential distribution of failure-free operating time at each of a finite number of machines and Exponential distribution of re-supply lead-time is developed. A graphical aid is presented which, for a given number of machines, indicates the range of the ratio {mean lead-time/mean failure-free operating time} for which a minimum S is required in order to satisfy a service level constraint on the service measure Pr[a spare is available at a machine stoppage due to part failure].     

Stochastic inventory system with lead time flexibility: offered by a manufacturer/transporter

This paper studies lead time flexibility in a two-stage continuous review supply chain in which the retailer uses the (R, Q) inventory system: when his inventory position reaches R, the retailer places orders with size Q to the manufacturer, who uses a transportation provider to deliver them with different lead time options. According to the contract, the manufacturer is able to expedite or postpone the delivery if the retailer makes such a request. Hence, the retailer has the flexibility to modify the lead time by using the most up-to-date demand information. The optimal lead time policy is found to be a threshold-type policy. The sensitivity analysis also shows that R is much more sensitive to the change of lead time than Q, and thus, the paper is primarily focused on finding optimal R. We also provide a cost approximation which yields unimodal cost in R. Furthermore, we analyze the order crossing problem and derive an upper bound for the probability of order crossing. Finally, we conduct an extensive sensitivity analysis to illustrate the effects of lead time flexibility on supply chain performance and discuss the managerial insights.     

Methods for Determining the Re-order Point of an (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) Ordering Policy when a Service Level is Specified

This paper deals with a periodic review inventory system. Methods are discussed for determining the re-order point s of an (s, S) order policy, when a certain service level is required. The results differ from those presented for a (Q, s) model which is usually considered in literature and implemented in practice. Methods are discussed for determining the re-order point of an (s, S) policy when demand is normal or gamma distributed. A numerical investigation demonstrates the applicability of the described methods. In particular, it is shown that these methods are superior to a formula that is implemented in many inventory control systems.     

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.     

Multi-period empty container repositioning with stochastic demand and lost sales

This paper considers repositioning empty containers between multi-ports over multi-periods with stochastic demand and lost sales. The objective is to minimize the total operating cost including container-holding cost, stockout cost, importing cost and exporting cost. First, we formulate the single-port case as an inventory problem over a finite horizon with stochastic import and export of empty containers. The optimal policy for period n is characterized by a pair of critical points (A _n , S _n ), that is, importing empty containers up to A _n when the number of empty containers in the port is fewer than A _n ; exporting empty containers down to S _n when the number of empty containers in the port is more than S _n ; and doing nothing, otherwise. A polynomial-time algorithm is developed to determine the two thresholds, that is, A _n and S _n , for each period. Next, we formulate the multi-port problem and determine a tight lower bound on the cost function. On the basis of the two-threshold optimal policy for a single port, a polynomial-time algorithm is developed to find an approximate repositioning policy for multi-ports. Simulation results show that the proposed approximate repositioning algorithm performs very effectively and efficiently.     

Periodic review inventory systems with a service level criterion

Full-cost inventory models are mostly studied in the literature, whereas service level constraints are more common to be observed in practical settings. In this paper, we consider periodic review inventory systems with service level restrictions. The control of such inventory systems is limited to (s, S)-type policies in the literature. To the best of our knowledge, we are the first authors to compare such policies with optimal replenishment policies, and illustrate an average cost difference of 0.64%. This justifies the use of these popular (s, S) policies in practice. Furthermore, we propose a new one-dimensional search procedure that is bounded to set the reorder level s and order-up-to level S, whereas the solution space is unbounded and two dimensional. Our heuristic procedure is guaranteed to satisfy the service level constraint and numerical experiments illustrate that it results in an average cost deviation of 1–2% compared with the best (s, S) policy. Consequently, it significantly outperforms all existing procedures from literature, both in service and costs.     

Analysis of an order-up-to-level policy for perishables with random issuing

Much of the research in perishable items inventory management has focused on the first-in-first-out issue process. However, motivated by the technical characteristics of the blood unit issue process, we model an order-up-to-level policy under periodic review setting with random issue of items from inventory. We provide empirical evidence in support of the random issuing assumption using real data on serial numbers of blood units issued from a blood bank. For general demand distribution we derive exact expressions for per period expected shortage, expected wastage and expected cost as functions of the policy parameters R (order-up-to-level) and T (review period). Since the exact model becomes computationally burdensome with increase in the number of periods of life of the perishable item an approximate model for the random issuing process is developed. The accuracy of the approximation is affirmed using simulation analysis. A gradient search-based heuristic is provided to identify the optimum policy parameters for the approximate model. A real life application of the model is demonstrated in determining the optimum frequency and order-up-to-level for blood collection at a blood bank.     

Reorder quantities for (<Emphasis Type="Italic">Q,R</Emphasis>) inventory models

The reorder level R and the reorder quantity Q are the parameters to be decided in a continuous review inventory policy. Their optimal values can be approached through iterative methods, but these are tedious and inconvenient for control routines. A frequent practice is to set Q as the economic reorder quantity and compute R accordingly. Yet this practice may introduce a substantial cost penalty. The paper re-arranges conventional theoretical expressions in order to facilitate the use of numerical approximations to help find a quasi-optimal solution. This approach is then applied to Gamma distributed demands showing that computations are straightforward.     

Ruin Probability in a Correlated Aggregate Claims Model with Common Poisson Shocks: Application to Reinsurance

This paper considers a correlated aggregate claims model with common Poisson shocks, which allows for dependence in n (n ≥ 2) classes of business across m (m ≥ 1) different types of stochastic events. The dependence structure between different claim numbers is connected with the thinning procedure. Under combination of quota-share and excess of loss reinsurance arrangements, we examine the properties of the proposed risk model. An upper bound for the ruin probability determined by the adjustment coefficient is established through martingale approach. We reduce the problem of optimal reinsurance strategy for maximizing the insurer’s adjustment coefficient and illustrate the results by numerical examples.     

Periodic review (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) inventory model with permissible delay in payments

This paper investigates the effect of permissible delay in payments on ordering policies in a periodic review (s, S) inventory model with stochastic demand. A new mathematical model is developed, which is an extension to that of Veinott and Wagner (Mngt Sci 1965; 11: 525) who applied renewal theory and stationary probabilistic analysis to determine the equivalent average cost per review period. The performance of the model is validated using a custom-built simulation programme. In addition, two distribution-free heuristic methods of reasonable accuracy develop approximate optimal policies for practical purposes based only on the mean and the standard deviation of the demand. Numerical examples are presented with results discussed.     

Two new heuristics for the <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">n</Emphasis>/0 queueing loss system with examples based on the two-phase Coxian distribution

In this paper, we introduce a new heuristic approach for the numerical analysis of queueing systems. In particular, we study the general, multi-server queueing loss system, the GI/G/n/0 queue, with an emphasis on the calculation of steady-state loss probabilities. Two new heuristics are developed, called the GM Heuristic and the MG Heuristic, both of which make use of an exact analysis of the corresponding single-server GI/G/1/0 queue. The GM Heuristic also uses an exact analysis of the GI/M/n/0 queue, while the MG Heuristic uses an exact analysis of the M/G/n/0 queue. Experimental results are based on the use of two-phase Coxian distributions for both the inter-arrival time and the service time; these include an error analysis for each heuristic and the derivation of experimental probability bounds for the loss probability. For the class of problems studied, it is concluded that there are likely to be many situations where the accuracy of the GM Heuristic is adequate for practical purposes. Methods are also developed for combining the GM and MG Heuristics. In some cases, this leads to approximations that are significantly more accurate than those obtained by the individual heuristics.     

Exact fill rates for (<Emphasis Type="Italic">R,s, S</Emphasis>) inventory control with gamma distributed demand

For the familiar (R, s, S) inventory control system only approximate expressions exist for the fill rate, ie the fraction of demand that can be satisfied from stock. Best-known are the approximations derived from renewal theory, holding under specific conditions; in particular, S–s should be reasonably large. Here, an exact expression for the fill rate is derived, holding generally in the situation that demand has a gamma distribution with known integer-valued shape parameter, while lead time is constant. These exact results allow a check of the renewal theory based approximations. In addition, an extremely fast simulation program was written, obviously holding for general shape parameter values.     

New heuristics for flow shop problem to minimize makespan

This paper investigates the flow shop problem with the objective to minimizemakespan. New algorithms are designed: one is off-line heuristic, Single JobFirst, for problemF _m ∥C _max ;and the other is on-line heuristic, Dynamic Single Job First (DSJF), for problemF _m |r _i |C _max and its on-line version. It is proved that the two heuristics are asymptoticallyoptimal when the size of the problem is large enough. In addition, theasymptotical optimality of First-Come, First-Served manner is obtained as abyproduct of the asymptotical analysis of DSJF heuristic. At the end of thepaper, a new lower bound with performance guarantee is provided for problemF _m ∥C _max , andcomputational experiments show the effectiveness of these heuristicalgorithms.