Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s,S) model with a maximum issue quantity restriction

The analysis of optimal inventory replenishment policies for items having lumpy demand patterns is difficult, and has not been studied extensively although these items constitute an appreciable portion of inventory populations in parts and supplies types of stockholdings. This paper studies the control of an inventory item when the demand is lumpy. A continuous review (s,S) policy with a maximum issue quantity restriction and with the possibility of opportunistic replenishment is proposed to avoid the stock of these items being depleted unduly when all the customer orders are satisfied from the available inventory and to reduce ordering cost by coordinating inventory replenishments. The nature of the customer demands is approximated by a compound Poisson distribution. When a customer order arrives, if the order size is greater than the maximum issue quantity w, the order is satisfied by placing a special replenishment order rather than from the available stock directly. In addition, if the current inventory position is equal to or below a critical level A when such an order arrives, an opportunistic replenishment order which combines the special replenishment order and the regular replenishment order will be placed, in order to satisfy the customer's demand and to bring the inventory position to S. In this paper, the properties of the cost function of such an inventory system with respect to the control parameters s, S and A are analysed in detail. An algorithm is developed to determine the global optimal values of the control parameters. Indeed, the incorporation of the maximum issue quantity and opportunistic replenishment into the (s,S) policy reduces the total operating cost of the inventory system.     

Inventory control in multi-channel retail

In recent years multi-channel retail systems have received increasing interest. Partly due to growing online business that serves as a second sales channel for many firms, offering channel specific prices has become a common form of revenue management. We analyze conditions for known inventory control policies to be optimal in presence of two different sales channels. We propose a single item lost sales model with a lead time of zero, periodic review and nonlinear non-stationary cost components without rationing to realistically represent a typical web-based retail scenario. We analyze three variants of the model with different arrival processes: demand not following any particular distribution, Poisson distributed demand and a batch arrival process where demand follows a Pòlya frequency type distribution. We show that without further assumptions on the arrival process, relatively strict conditions must be imposed on the penalty cost in order to achieve optimality of the base stock policy. We also show that for a Poisson arrival process with fixed ordering costs the model with two sales channels can be transformed into the well known model with a single channel where mild conditions yield optimality of an (s, S) policy. Conditions for optimality of the base stock and (s, S) policy for the batch arrival process with and without fixed ordering costs, respectively, are presented together with a proof that the batch arrival process provides valid upper and lower bounds for the optimal value function.     

Inventory control with indivisible units of stock transfer

We consider an infinite horizon, single item inventory model with backorders and a fixed lead time. Demand is stationary stochastic and review is periodic. Inventory may only be replenished in multiples of a fixed package size q but demands may be of any size. Ordering costs are linear and combined holding and shortage costs can be expressed as a convex function of the inventory position. The control policy is defined as (s, S, q), where an order is placed if the inventory position falls to or below s and the order size is the largest multiple of q which results in the inventory position not exceeding S. The parameters s and S are restricted to be multiples of q. The objective is to find the control policy that minimizes the long run average cost per unit time. The optimal solution procedure requires renewal theory and a structured search. Fortunately, a heuristic based on the ‘quantized ordering’ approach of Zheng and Chen provides solutions that are near optimal over a broad range of parameter values.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

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.     

Lost-sales inventory systems with a service level criterion

Competitive retail environments are characterized by service levels and lost sales in case of excess demand. We contribute to research on lost-sales models with a service level criterion in multiple ways. First, we study the optimal replenishment policy for this type of inventory system as well as base-stock policies and (R, s, S) policies. Furthermore, we derive lower and upper bounds on the order-up-to level, and we propose efficient approximation procedures to determine the order-up-to level. The procedures find values of the inventory control variables that are close to the best (R, s, S) policy and comply to the service level restriction for most of the instances, with an average cost increase of 2.3% and 1.2% for the case without and with fixed order costs, respectively.     

Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity – the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.     

Failure replacement and preventive maintenance spare parts ordering policy

This paper addresses inventory policy for spare parts, when demand for the spare parts arises due to regularly scheduled preventive maintenance, as well as random failure of units in service. A stochastic dynamic programming model is used to characterize an ordering policy which addresses both sources of demand in a unified manner. The optimal policy has the form (s(k),S(k)), where k is the number of periods until the next scheduled preventive maintenance operation. The nature of the (s(k),S(k)) policy is characterized through numeric evaluation. The efficiency of the optimal policy is evaluated, relative to a simpler policy which addresses the failure replacement and preventive maintenance demands with separate ordering policies.     

Dynamic inventory and pricing policy in a capacitated stochastic inventory system with fixed ordering cost

We study a dynamic inventory and pricing optimization problem in a periodic review inventory system with setup cost and finite ordering capacity in each period. We show that the optimal inventory control is characterized by an (s,s^′,p) policy in four regions of the starting inventory level.     

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.     

Coordinating inventory control and pricing strategies: The continuous review model

We analyze an infinite horizon, single product, continuous review model in which pricing and inventory decisions are made simultaneously and ordering cost includes a fixed cost. We show that there exists a stationary (s,S) inventory policy maximizing the expected discounted or expected average profit under general conditions.     

On exhibiting inventory systems with Erlangian lifetimes under renewal demands

A continuous revies (s, S) inventory system with renewal demand in which one item is put into operation as an exhibiting piece is analyzed. The lifetime of any operating unit has Erlangian distribution, and on failure is replaced by another one from the stock and the failed item is disposed of. Replenishment of stock is instantaneous. The transient and stationary values of inventory level distribution and the mean reorder rate are obtained using the techniques of semi-regenerative processes. Decision rules for optimums andS that minimize the long-run expected cost rate are derived. The solution for a dual model with the distribution of lifetimes and inter-demand times interchanged is also given.     

On a stochastic inventory model with a generalized holding costs

This paper is concerned with finding the optimal replenishment policy for an inventory model that minimizes the total expected discounted costs over an infinite planning horizon. The demand is assumed to be driven by a Brownian motion with drift and the holding costs (inventory and shortages) are assumed to take some general form. This generalizes the earlier work where holding costs were assumed linear. It turns out that problem of finding the optimal replenishment schedule reduces to the problem of solving a Quasi-Variational Inequality Problem (QVI). This QVI is then shown to lead to an (s, S) policy, where s and S are determined uniquely as a solution of some algebraic equations.     

Inventory models with minimal service level constraints

This paper investigates inventory models in which the stockout cost is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met in every period. The minimal service level approach has the virtue of simplifying the computation of an optimal ordering policy, because the optimal reorder level is solely determined by the minimal SLC and demand distributions. It is found that above a certain “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to level policy, which is independent on the cost parameters. This shows the minimal SLC models to be qualitatively different from their shortage cost counterparts. We also demonstrate that the “imputed shortage cost” transforming a minimal SLC model to a shortage cost model does not generally exist. The minimal SLC approach is extended to models with negligible set-up costs. The optimality of myopic base-stock policies is established under mild conditions.     

A dyadic age-replacement policy for a periodically inspected equipment item subject to random deterioration

A periodic review replacement system is considered. The amount of deterioration over successive periods forms a sequence of i.i.d. random variables. A replacement policy of the dyadic type is in effect whereby the used equipment item is discarded and immediately replaced by a new identical equipment item if at the end of a period the old equipment has service aged by an amount in excess of S or has been in operation for exactly N periods whichever comes first. Using a theorem on renewal reward processes, an expression for the total steady-state expected cost per period is derived, consisting of a fixed replacement cost and a linear cost of operation. Optimal values of S and N that minimize this steady state cost are computed for a few numerical examples, when the service aging per period has a gamma distribution.     

A condition-based replacement and spare provisioning policy for deteriorating systems with uncertain deterioration to failure

A new policy, referred to as the condition-based replacement and spare provisioning policy, is presented for deteriorating systems with a number of identical units. It combines the condition-based replacement policy with periodical inspections and the (S,s$S\text{,}s$) type inventory policy, noted as the (T,S,s,_Lp$T\text{,}S\text{,}s\text{,}{L}_{\text{p}}$) policy, where T is the inspection interval, S is the maximum stock level, s   is the reorder level, and _Lp$L_{\text{p}}$ is the preventive replacement threshold for the deterioration levels of units. The deterioration level of each unit in the system can be described by a scalar random variable, which is continuous and increasing monotonically. Furthermore, the deterioration level just when the unit failure occurs, termed deterioration to failure, is uncertain. Therefore, the condition-based reliability is proposed in order to characterize various and uncertain deterioration levels when unit failure occurs. A simulation model is developed for the system operation under the proposed condition-based replacement and spare provisioning policy. Thus, via the simulation method and the genetic algorithm, the decision variables T, S, s  , and _Lp$L_{\text{p}}$ can be jointly optimized for minimizing the cost rate. A case study is given, showing the procedure of applying the proposed policy and the condition-based reliability methodology to optimizing the maintenance scheme of haul truck motors at a mine site based on oil inspections, and proving beneficial for plant maintenance managers to reduce maintenance cost.     

Modelling and computing (Rn, Sn) policies for inventory systems with non-stationary stochastic demand

This paper addresses the single-item, non-stationary stochastic demand inventory control problem under the non-stationary (R, S) policy. In non-stationary (R, S) policies two sets of control parameters—the review intervals, which are not necessarily equal, and the order-up-to-levels for replenishment periods—are fixed at the beginning of the planning horizon to minimize the expected total cost. It is assumed that the total cost is comprised of fixed ordering costs and proportional direct item, inventory holding and shortage costs. With the common assumption that the actual demand per period is a normally distributed random variable about some forecast value, a certainty equivalent mixed integer linear programming model is developed for computing policy parameters. The model is obtained by means of a piecewise linear approximation to the non-linear terms in the cost function. Numerical examples are provided.     

An (s,S) random lifetime inventory model with a positive lead time

Positive lead times substantially complicate the modeling and analysis of inventory systems with finite shelf lifetimes and they have not been sufficiently addressed in the existing literature. In this paper, we analyze an (s, S) continuous review model with a positive lead time. We assume an exponential lifetime and an exponential lead time. Matrix-geometric solutions can be obtained for the steady state probability distribution of the inventory level. We then derive the total expected cost function. We carry out numerical studies and gain insights to the selection of system parameters. The findings on the impact of a positive lead time on the optimal reorder point and reorder batch size will be useful in developing strategies in handling inventory problems with finite lifetimes and positive lead times.     

On a stochastic demand jump inventory model

This paper considers a Quasi-Variational Inequality (QVI) arising from a stochastic demand jump inventory model in a continuous review setting with a fixed ordering cost and where demand is made up of a deterministic part (which is a function of the stock level) punctuated by random jumps. Under some restrictions on the parameters, a solution to the QVI is found which corresponds to an (s,S) policy.