Computing optimal ( s,S ) policies for inventory systems with a cut-off transaction size and the option of joint replenishment

This paper presents a mathematical model that is developed for the synthesis of optimal replenishment policies for items exhibiting lumpy demand patterns. In order to avoid the disruption of the inventory system, a maximum issue quantity restriction is incorporated into the inventory control policy. In doing so, customer demands with a size exceeding w units will be filtered out of the inventory system, but satisfied by placing a special replenishment order. The rest of the customer demands will be met from stock. The control discipline is the continuous review ( s, S ) inventory policy and the nature of the demands is approximated by a discrete compound Poisson distribution. To further reduce the total annual ordering cost, specification is made such that when the current inventory position is below a critical level, A , at the time when a customer demand with a size exceeding w units arrives, a joint replenishment is placed that will not only initiate a direct shipment to the customer, but will also raise the inventory position to S . An algorithm is developed to determine the global optimal values of the control parameters, s and S for given w and A . A numerical example is used to illustrate the methodology, and the results obtained clearly show that a better cost performance can be obtained with the incorporation of both a maximum issue quantity restriction and the option of joint replenishment into the inventory control policy.     

A joint replenishment policy with individual control and constant size orders

We consider inventory systems with multiple items under stochastic demand and jointly incurred order setup costs. The problem is to determine the replenishment policy that minimises the total expected ordering, inventory holding, and backordering costs–the so-called stochastic joint replenishment problem. In particular, we study the settings in which order setup costs reflect the transportation costs and have a step-wise cost structure, each step corresponding to an additional transportation vehicle. For this setting, we propose a new policy that we call the (s, 𝒬) policy, under which a replenishment order of constant size 𝒬 is triggered whenever the inventory position of one of the items drops to its reorder point s. The replenishment order is allocated to multiple items so that the inventory positions are equalised as much as possible. The policy is designed for settings in which backorder and setup costs are high, as it allows the items to independently trigger replenishment orders and fully exploits the economies of scale by consistently ordering the same quantity. A numerical study is conducted to show that the proposed (s, 𝒬) policy outperforms the well-known (𝒬, S) policy when backorder costs are high and lead times are small.     

Coordinated replenishment systems with discount opportunities

In many practical situations, coordination of replenishment orders for a family of items can lead to considerable cost savings. A well-known class of strategies for the case where cost savings are due to reduced joint ordering costs is the class of can-order strategies. However, these strategies, which are simple to implement in practice, do not take discount possibilities into account. We propose a method to incorporate discounts in the framework of can-order strategies. A continuous review multi-item inventory system is considered with independent compound Poisson demand processes for each of the individual items. Discounts are offered by the supplier as a percentage of the total dollar value whenever this value exceeds a given threshold. Starting from the can-order strategy as a basic decision rule, we develop a simple heuristic to evaluate these discount opportunities. The performance of the can-order strategy with discount evaluation is compared with that of another class of discount evaluation rules as proposed by Miltenburg and Silver.     

Optimal inventory control of lumpy demand items using (s,S) policies with a maximum issue quantity restriction and opportunistic replenishments

This paper presents a modified (s, S) inventory model which describes the characteristics of an inventory system with lumpy demand items. A maximum issue quantity restriction of w units and a critical inventory position of A units are incorporated into the inventory control policy. Customer orders with demand sizes larger than the maximum issue quantity will be filtered out from the inventory system and satisfied by using special replenishment orders in order to avoid disruption to the inventory system. The option of opportunistic replenishments is introduced to further minimize the total replenishment cost. An opportunistic replenishment is initiated if the level of the current inventory position is equal to or below the critical level when a customer demand with a size exceeding the maximum issue quantity arrives, which does not only initiate a direct shipment to the customer, but also raises the inventory position to S. Two effective algorithms are developed to determine the optimal values of w, A, s and S simultaneously. The first algorithm is based on the branch-and-bound tree search technique, and the second one is based on the concept of genetic algorithms. Numerical examples are used to illustrate the effectiveness of the algorithms developed. The effects of changes in the cost and system parameters on the optimal inventory control policy are also studied by using sensitivity analysis.     

A stochastic model for joint inventory and outbound shipment decisions

In this paper, we consider a vendor realizing a sequence of random order arrivals in random sizes. The vendor has the autonomy to hold/consolidate small orders until an economical dispatch quantity accumulates. Consequently, the actual inventory requirements at the vendor are in part determined by the parameters of the shipment release policy in use. In this context, we investigate the impact of shipment consolidation on the expected long-run average cost by simultaneously computing the optimal order quantity for inventory replenishment at the vendor and the optimal dispatch quantity for outbound shipments. Since we consider the case where demand follows a general stochastic bulk arrival process, obtaining exact analytical expressions for some key operating characteristics of the cost function is intractable. Hence, we provide easy-to-compute approximations which enable efficient numerical solutions for the problem. We also investigate: (i) the cases where consolidated shipments are preferred over immediate deliveries; (ii) the sensitivity of optimal integrated policy variables to demand/cost parameters; (iii) the potential savings that can be obtained by shipment consolidation; and (iv) the tradeoffs between the waiting time induced by shipment consolidation and costs saved. Our results provide insights into the impact of outbound transportation operations on inventory replenishment decisions and outbound distribution system design. Moreover, numerical testing suggests that significant cost savings (up to 57%) are possible with shipment consolidation.     

RANDOM OPPORTUNITIES FOR REDUCED COST REPLENISHMENTS

In this paper we consider the situation where the typical assumptions underlying the EOQ model are appropriate for an item, in particular a reasonably level demand rate exists. However, opportunities for special replenishments at a reduced unit cost occur at random. The following type of 3-parameter policy is considered. When a non-special (regular) replenishment is made, an order-up-to-level, S₁;, is used. When a special opportunity arises we only take advantage of it if the current inventory is low enough (L₂ or lower) and we order enough to raise the inventory level to S₁. This situation was modelled earlier in the literature, but the solution procedure suggested involved a 3-dimensional search on the three control parameters, S₁,, L₂, and S₂. We develop an approximate solution method which is much easier to use and which performs excellently on a wide range of examples. The performance of an even more simple method, based on always taking advantage of a special opportunity to replenish at reduced unit cost (i.e., setting L% = S₂), is shown to be highly dependent upon the values of the five uncontrollable parameters of the problem.     

Policy and cost approximations of two-echelon distribution systems with a procurement cost at the higher echelon

This paper investigates a two-echelon (warehouse-retailer) inventory system with stochastic demand and a pull system of inventory allocation. We assume that ordering costs are charged at the warehouse for procuring the item from a supplier. However, the internal costs of ordering the item from the warehouse by the retailers are considered negligible. For this problem, the lower echelon uses a single critical number, an order-up-to-level policy, whereas a (s, S) inventory system is followed at the upper echelon. We develop a cost model for this problem and provide a simple algorithm for estimating the optimal policy. Simulation is used to test the accuracy of the model.     

AN ITERATIVE ANALOG INVENTORY POLTCY COST MODEL

This article describes the use of an analog computer, operating in the iterative mode, to perform simulations of inventory policy situations. The inventory policies studied are the re-order level, re-order cycle and (s, S) policies, and within each of these inventory policies a replenishment order policy of either a fixed or variable type can be operated. All policies are subject to stochastic sized customer demand and lead-time durations and a cost model is developed to compare the operating costs of the various policies under these conditions. This cost model is used further to investigate in some detail the self-adaptability and the optimal character of the (s, S) policy.     

Value of information exchange and synchronization in a multi-tier supply chain

We investigate the value of various information exchange mechanisms in a four-echelon supply chain under a material requirements planning framework. In the absence of any information sharing, each echelon would develop its own forecasts and plan its inventories based on the history of actual demand from its downstream customer (or echelon). Through a simulation study, we compare this policy with policies where each echelon has access to (i) the end-user demand history and (ii) the planned order schedule of the downstream echelon. Among all the demand information exchange mechanisms, planning inventories based on the planned downstream order schedules resulted in the lowest average inventory level for the entire supply chain. However, use of end-user demand history to forecast and plan inventories at all echelons resulted in the lowest total cost. In addition to the information exchange mechanisms, a simple synchronized replenishment system was considered and evaluated in the study. In the synchronized system, the retailer determines a fixed order interval and the upper echelons replenish only at integer multiples of this interval. The study found that synchronized inventory replenishments among the echelons, even without any exchange of demand information, can bring about more benefits and cost reduction than any of the information exchange mechanisms.     

Inventory replenishment policies for deteriorating items in a declining market

In this paper two mathematical models are developed for an inventory system in which the units are deteriorating at a constant rate and the demand rate decreases negative exponentially. In the first model it is assumed that replenishment orders are placed at equal intervals while in the second model the replenishment times are also variables and hence there is no need for the replenishment orders to be placed at equal intervals. Optimal replenishment policies are determined for both cases. A numerical example is used to illustrate the theory. Computational results indicate that the constant replenishment period policy leads to a higher total cost.     

The supplier–buyer integrated production-inventory model with random yield

This paper revisits the traditional supplier–buyer integrated production-inventory model which deals with the problem of a manufacturer (supplier) supplying a product to a retailer (buyer) serving the consumer market with constant stationary demand. The product is manufactured in batches at a finite rate. The supplier's production batch is depleted by the buyer's replenishment orders at periodic intervals. The buyer's inventory is then consumed by the market demand at a fixed rate. The problem is the simultaneous computation of the manufacturer's production lot-size and the buyer's replenishment order quantity, i.e. the integrated production-inventory policy parameters. The key characteristic considered in this paper is that the manufacturing process is imperfect, and, hence, there are defective items in each production lot. As a result, each replenishment order shipped to the buyer includes defective products and the non-defective percentage in each such shipment is random. Considering the case where the supplier replenishes the buyer via equal-sized shipments, we develop an analytical expression of the total expected cost for the supplier–buyer system under consideration, with and without a considerable inspection time. We first examine the case where the inspection time is negligible, and then we present a generalisation to consider the inspection time explicitly. Our goal is to model the impact of random yield on the system performance. Our findings are useful for computing the integrated production-inventory policy parameters while considering the supply uncertainty due to an imperfect manufacturing process. Through numerical examples, we quantify the impact of supply with random yield on the system performance and illustrate its relationship with the demand and production rate.     

Periodic-review inventory systems with random yield and demand: Bounds and heuristics

We revisit the infinite-horizon decision problem of a single-stage single-item periodic-review inventory system under uncertain yield and demand. It is known that under some mild conditions the optimal replenishment policy is of the threshold type: an order is placed if and only if the starting inventory is below a threshold value. Although the structure of the optimal policy is well known, there has been little discussion about the optimal order quantities and the order threshold. In this paper, we construct upper and lower bounds for the optimal threshold value and the optimal order quantities through solving one-period problems with different cost parameters. These bounds provide interesting insights into the impact of yield uncertainty on the optimal policy. Heuristics are developed based on these bounds. Detailed computational studies show that, under some conditions, the performance of the heuristics is very close to that of the optimal solution and better than that of existing heuristics in the literature.     

Optimal inventory policies for uniform replenishment systems with time-dependent demand

This study develops an analysis of lot size inventory systems where the replenishment rate is uniform and demand follows a power demand pattern. Shortages are not allowed. Holding cost, replenishing cost and purchasing cost are considered in inventory system control. The objective of the study is to find the economic production quantity that minimises total inventory cost per unit of time. We conclude that optimal inventory policies depend on the demand pattern index chosen to represent customer demand. Theoretical results are illustrated with a business case study. A sensitivity analysis is proposed to describe the optimal policy behaviour.     

Optimal Inventory Policy for Systems with Demand Cancellations

We analyze a periodic review inventory model in which all demands have to be reserved with a one-period leadtime. During the reservation period, some of the reservations may be cancelled. As a result, the reservation information is not perfectly reliable. Our objective is to determine the optimal replenishment policy for this inventory model. We show that the cost function is convex and the optimal policy is a "reorder-up-to" type of policy. This policy stipulates that unless the initial on-hand inventory is lower than the reorder-up-to level, an order should be made to raise the inventory level to the reorder-up-to level. Furthermore, this reorder-up-to level is a function of the amount of reserved demand. The single- and multiple-period cases are studied. Finally, we consider the infinite-horizon case and show that results from the finite-horizon case converge to the infinite-horizon case. The expression for solving the optimal policy parameter is also derived.     

Optimal stocking strategies for inventory mechanism with a stochastic short-term price discount and partial backordering

Price discount is an important research topic in the field of inventory management. The existing research on this topic mainly considers fixed price discount, but ignores the situation in which stochastic short-term price discount may be involved. In this paper, we study an inventory problem considering stochastic short-term price discount and partial backordering. To address this problem, we propose an optimal replenishment and stocking model to maximise the retailers' profit. After that, a cost–benefit analysis-based heuristic method for solving the developed model is presented by considering two scenarios depending on whether a replenishment point belongs to a discount period or not. Furthermore, an algorithm is provided to elicit an optimal ordering policy from multiple solutions derived from the given heuristic solution method. Finally, a real case is offered to demonstrate the application of the proposed model, followed by a sensitivity analysis. The results indicate that a retailer can identify the optimal replenishment policy with the aim of achieving maximal profit in situations where stochastic short-term price discount and partial backordering are considered for certain inventory problems at hand. In addition, sensitivity analysis illustrates a fact that different values of the introduced parameters may influence the optimal replenishment policy.     

Optimal joint replenishment policy for multiple non-instantaneous deteriorating items

In this paper, we deal with the problem of determining the optimal economic operating policy when a number of non-instantaneous deteriorating items are jointly replenished. We establish a multi-item joint replenishment model for non-instantaneous deteriorating items under constant demand rate allowing full backlogging. This problem is challenging, in particular, the cost function is a piecewise function with exponential parts, which makes the problem more complicated. To solve this problem, an approximation method is used to simplify the objective function and a bound-based heuristic algorithm is developed to solve the model. Numerical examples illustrate the effectiveness of the proposed method and the quality of the approximation. Experimental results on a real-life case study show that the proposed model can achieve substantial cost savings compared to the individual replenishment policy for non-instantaneous deteriorating items. Furthermore, sensitivity analysis of key parameters is carried out and the implications are discussed in detail.     

Applying machine learning to the dynamic selection of replenishment policies in fast-changing supply chain environments

Firms currently operate in highly competitive scenarios, where the environmental conditions evolve over time. Many factors intervene simultaneously and their hard-to-interpret interactions throughout the supply chain greatly complicate decision-making. The complexity clearly manifests itself in the field of inventory management, in which determining the optimal replenishment rule often becomes an intractable problem. This paper applies machine learning to help managers understand these complex scenarios and better manage the inventory flow. Building on a dynamic framework, we employ an inductive learning algorithm for setting the most appropriate replenishment policy over time by reacting to the environmental changes. This approach proves to be effective in a three-echelon supply chain where the scenario is defined by seven variables (cost structure, demand variability, three lead times, and two partners’ inventory policy). Considering four alternatives, the algorithm determines the best replenishment rule around 88% of the time. This leads to a noticeable reduction of operating costs against static alternatives. Interestingly, we observe that the nodes are much more sensitive to inventory decisions in the lower echelons than in the upper echelons of the supply chain.     

Base stock versus WIP cap in single-stage make-to-stock production–inventory systems

A single-stage production-inventory system produces parts in a make-to-stock mode, and unsatisfied demand is backordered. The system operates under a so-called base stock with WIP cap replenishment policy, which works as follows. Whenever the Work-In-Process (WIP) plus finished goods inventory falls below a specified level, called base stock, a replenishment order for the production of a new part is issued. If the WIP inventory is below a different specified level, called WIP cap, the order goes through and a new part is released for production; otherwise, the order is put on hold until the WIP inventory drops below the WIP cap. First, it is shown that the optimal base stock that minimizes the long-run, average, inventory holding cost for a given minimum customer service level, is a non-increasing function of the WIP cap that reaches a minimum value, called minimum optimal base stock, at a finite WIP cap value, called critical WIP cap. Then, it is shown that the optimal WIP cap is less than or equal to the critical WIP cap and therefore the optima! base stock is greater than or equal to the minimum optimal base stock. More interestingly, however, it is conjectured that the optimal WIP cap is in fact exactly equal to the critical WIP cap and therefore the optimal base stock is exactly equal to the minimum optimal base stock. The minimum optimal base stock is none other than the optimal base slock of the same system operating under a classical base stock policy (with no WIP cap). Finally, the optimal parameters of a system operating under a base stock with WIP cap policy are related to the optimal parameter of the same system operating under a make-to-stock CONWIP policy.     

Inventory control of defective products when the demand is partially captive

In this paper a mathematical model is developed for an inventory system in which the number of units of acceptable quality in a replenishment lot is uncertain and the demand is partially captive. It is assumed that the fraction of the demand during the stockout period which can be backordered is a random variable whose probability distribution is known. The optimal replenishment policy is synthesized for such a system. A numerical example is used to illustrate the theory. The results indicate that the optimal replenishment policy is sensitive to the nature of the demand during the stockout period.     

Frequency approach for treating capacity-constrained multilevel production

An idea about how to tackle and solve the difficult problem of determining economic order quantities in capacity-constrained multilevel production is presented. The idea is based on frequencies during a variable time interval as variables instead of explicit order quantities. The presented heuristic model handles assumed constant demand for several items and calculates fixed-order quantities assumed to be possible to repeat in a cyclic pattern. In the model, every item is supposed to be manufactured at least once per year, or during another maximum period, to stop order quantities that are too large and therefore hinder inventories that are exceedingly large and/or obsolete. During a time interval, it was decided how many times each item should be produced. The paper starts with a common frequency for all items, i.e. the highest frequency possible for all items during this maximum time interval considering the capacity restrictions of the different machines. Thereafter, the frequency for items with high set-up costs compared with the inventory holding costs is reduced if it decreases the total cost of set-ups and inventory holding. Only frequencies in multiples of two are used to make a cycle policy easier to establish, to prevent remnant stocks and to make the used cost approximation fit more accurately.