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.     

Multi-item two-echelon spare parts inventory control problem with batch ordering in the central warehouse under compound Poisson demand

We consider a multi-item two-echelon spare part inventory system in which the central warehouse operates under an (nQ,?R) policy and the local warehouses implement order-up-to S policy, each facing a compound Poisson demand. The objective is to find the policy parameters minimizing expected system-wide inventory holding and fixed ordering costs subject to an aggregate mean response time constraint at each warehouse. In this paper, we propose four alternative approximations for the steady state performance of the system; and extend a heuristic and a lower bound proposed under Poisson demand assumption to the compound Poisson setting. In a computational study, we show that the performances of the approximations, the heuristic, and the lower bound are quite satisfactory; and the relative cost saving of setting an aggregate service level rather than individually for each part is quite high.     

<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 type queueing-inventory systems with postponed work,reservation, cancellation and common life time

In this paper we analyze two single server queueing-inventory systems in which items in the inventory have a random common life time. On realization of common life time, all customers in the system are flushed out. Subsequently the inventory reaches its maximum level S through a (positive lead time) replenishment for the next cycle which follows an exponential distribution. Through cancellation of purchases, inventory gets added until their expiry time; where cancellation time follows exponential distribution. Customers arrive according to a Poisson process and service time is exponentially distributed. On arrival if a customer finds the server busy, then he joins a buffer of varying size. If there is no inventory, the arriving customer first try to queue up in a finite waiting room of capacity K. Finding that at full, he joins a pool of infinite capacity with probability γ (0 < γ < 1); else it is lost to the system forever. We discuss two models based on ‘transfer’ of customers from the pool to the waiting room / buffer. In Model 1 when, at a service completion epoch the waiting room size drops to preassigned number L ? 1 (1 < L < K) or below, a customer is transferred from pool to waiting room with probability p (0 < p < 1) and positioned as the last among the waiting customers. If at a departure epoch the waiting room turns out to be empty and there is at least one customer in the pool, then the one ahead of all waiting in the pool gets transferred to the waiting room with probability one. We introduce a totally different transfer mechanism in Model 2: when at a service completion epoch, the server turns idle with at least one item in the inventory, the pooled customer is immediately taken for service. At the time of a cancellation if the server is idle with none, one or more customers in the waiting room, then the head of the pooled customer go to the buffer directly for service. Also we assume that no customer joins the system when there is no item in the inventory. Several system performance measures are obtained. A cost function is discussed for each model and some numerical illustrations are presented. Finally a comparison of the two models are made.     

Inventory management of spare parts in an energy company

We address the problem of how to determine control parameters for the inventory of spare parts of an energy company. The prevailing policy is based on an (s, S) system subject to a fill rate constraint. The parameters are decided based mainly on the expert judgment of the planners at different plants. The company is pursuing to conform all planners to the same approach, and to be more cost efficient. Our work focuses on supporting these goals. We test seven demand models using real-world data for about 21?000 items. We find that significant differences in cost and service level may appear from using one or another model. We propose a decision rule to select an appropriate model. Our approach allows us to recommend control parameters for 97.9% of the items. We also explore the impact of pooling inventory for different demand sources and the inaccuracy arising from duplicate item codes.     

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.     

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.     

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.     

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].     

Distribution Free Procedures for Some Inventory Models

Stochastic inventory models, such as continuous review models and periodic review models, require information on the lead time demand. However, information about the form of the probability distribution of the lead time demand is often limited in practice. We relax the assumption that the cumulative distribution function, say F, of the lead time demand is completely known and merely assume that the first two moments of F are known and finte. The minmax distribution free approach for the inventory model consists of finding the most unfavourable distribution for each decision variable and then minimizing over the decision variable. We solve both the continuous review model and the periodic review model with a mixture of backorders and lost sales using the minmax distribution free approach.     

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.     

Optimal retailer's ordering policies in the EOQ model under trade credit financing

The main purpose of this note is to modify the assumption of the trade credit policy in previously published results to reflect the real-life situations. All previously published models implicitly assumed that the supplier would offer the retailer a delay period, but the retailer would not offer the trade credit period to his/her customer. In most business transactions, this assumption is debatable. In this note, we assume that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer's replenishment model. Furthermore, we assume that the retailer's trade credit period offered by supplier M is not shorter than the customer's trade credit period offered by retailer N(M?N). Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal ordering policies. Then a theorem is developed to determine efficiently the optimal ordering policies for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this note.     

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.     

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.     

A note on an <Emphasis Type="Italic">L</Emphasis>-approach for solving the manufacturer's pallet loading problem

An L-approach for packing (l,?w)-rectangles into an (L,?W)-rectangle was introduced in an earlier work by Lins, Lins and Morabito. They conjecture that the L-approach is exact and point out its runtime requirements as the main drawback. In this note it is shown that, by simply using a different data structure, the runtime is considerably reduced in spite of larger (but affordable) memory requirements. This reduction is important for practical purposes since it makes the algorithm much more acceptable for supporting actual decisions in pallet loading. Intensive numerical experiments showing the efficiency and effectiveness of the algorithm are presented.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

Packing circles in the smallest circle: an adaptive hybrid algorithm

The circular packing problem (CPP) consists of packing n circles C _i of known radii r _i , i∈N={1,?…,?n}, into the smallest containing circle ?. The objective is to determine the coordinates (x _i ,?y _i ) of the centre of C _i , i∈N, as well as the radius r and centre (x,?y) of ?. CPP, which is a variant of the two-dimensional open-dimension problem, is NP hard. This paper presents an adaptive algorithm that incorporates nested partitioning within a tabu search and applies some diversification strategies to obtain a (near) global optimum. The tabu search is to identify the n circles’ ordering, whereas the nested partitioning is to determine the n circles’ positions that yield the smallest r. The computational results show the efficiency of the proposed algorithm.     

Some replenishment rules considering payment periods and risk of outdating

This paper is devoted to the study of the effects of payment terms on the replenishment strategy. We use a discounting approach with Poisson demand, a linear inventory carrying cost and a very short replenishment lead time. The case of repeated identical payments at the end of a specified period of t days, as well as the case of a one time longer payment period of u days, are taken into consideration. In both situations we are able to develop rules to determine the appropriate replenishment quantities. We show how these rules change depending on whether the goods provided are subject to oudating or not. Numerical and graphical illustrations are provided.     

MULTITRANSITIVITY OF CALOGERO-MOSER SPACES

Let G be the group of unimodular automorphisms of a free associative ?-algebra on two generators. A theorem of G. Wilson and the first author [BW] asserts that the natural action of G on the Calogero-Moser spaces C _n is transitive for all n ? ?. We extend this result in two ways: first, we prove that the action of G on C _n is doubly transitive, meaning that G acts transitively on the configuration space of ordered pairs of distinct points in C _n ; second, we prove that the diagonal action of G on \( {C}_{n_1}\times {C}_{n_2}\times \cdots \times {C}_{n_m} \) is transitive provided n ₁,?n ₂,?…,?n _m are pairwise distinct numbers.     

The modular group and words in its two generators<Superscript>*</Superscript>

Consider the full modular group PSL₂(?) with presentation 〈U,?S|U ³,?S ²〉. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper may be considered as a necessary appendix), we are led to the following natural question. Some words in the alphabet {U, S} are equal to the unity; for example, USU ³ SU ² is such a word of length 8, and USU ³ SUSU ³ S ³ U is such a word of length 15. We consider the following integer sequence. For each n ∈ ?₀, let t(n) be the number of words in the alphabet {U, S} that equal the identity in the group. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over ?(x) of degree 3. As an interesting generalization, we formulate the problem of describing all algebraic functions with a Fermat property.