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.     

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.     

Ranking of generalised multi-stage KANBAN policies

We consider KANBAN type batch-ordering policies in a multi-stage production–inventory system. KANBAN policies and various modifications of such policies may be interpreted as installation stock (Q, r)-policies with constraints on the number of outstanding orders. If the same constraints are applied also to MRP and echelon stock (Q, r)-policies, the ranking lists established by Axsäter and Rosling (Axsäter, S., Rosling, K., 1993. Management Science 39, 1274–1280; Axsäter, S., Rosling, K., 1994. European Journal of Operational Research 75, 405–412) for such policies remain valid. The result is based on new definitions of the inventory positions, and suggests new classes of generalised KANBAN type batch-ordering policies with potentially improved performance.     

Continuous-review inventory problem with random supply interruptions

This paper considers a continuous-review stochastic inventory problem with random demand and random lead-time where supply may be disrupted due to machine breakdowns, strikes or other randomly occurring events. The supplier availability is modelled as a semi-Markov process (more specifically, as an alternating renewal process). The standard (q, r) policy is used when the supplier is available (ON), i.e., when the inventory position reaches the reorder point r, q units are ordered to raise the inventory position to the target level of R = q + r. The form of the policy changes when the supplier becomes unavailable (OFF) in which case orders cannot be placed when the reorder point r is reached. However, as soon as the supplier becomes available again one orders enough to bring the inventory position up to the target level of R. The regenerative cycles are identified by observing the inventory position process. We construct the average cost per time objective function using the renewal reward theorem. It is assumed that the duration of the ON period is E_k (i.e., k-stage Erlangian) and the OFF period is general. In analogy with queuing notation we call this an E_k/G system. By employing the ‘method of stages’, we obtain a problem with a larger state space for the ON/OFF stochastic process; but the resulting ON process can now be analyzed using Markovian techniques. For asymptotic values of q, the objective function assumes a particularly simple form which is shown to be convex under mild restrictions on the density functions of demand. Numerical examples illustrate the results.     

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.     

A Quality Control Approach for Monitoring Inventory Stock Levels

In this paper we develop a new approach to monitor the accuracy of an inventory management system. A recorded stock level is considered accurate when the recorded level agrees with the actual stock level, otherwise there is an error. In practice, management relies on methods to measure or assure inventory accuracy not necessarily developed for this purpose. Our methodology is based on the average absolute relative difference as a simple analytical measure for inventory accuracy (AC _N ). The approach captures the status of accuracy in an inventory and allows for greater understanding of what affects inaccuracy since the theoretical measure of accuracy is composed of several parameters representing the incidence and proportion of both overstock and understock. The implementation of the methodology is constrained because complete inspection of the inventory is very expensive in most situations, so we develop the sample analogue of the accuracy measure (AC _n ) and discuss sampling strategies. The accuracy of the inventory system is monitored by incorporating AC _n into a univariate control chart.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

Non-stationary inventory position processes

This paper investigates single-product non-stationary inventory problems associated with non-stationary demand processes, backorders allowed, stochastically independent inter-review-period demands, and review periods not necessarily of equal length. Sufficient conditions are obtained for the convergence in distribution of inventory positions in 〈R, r, T〉 inventory systems; in particular, the uniform long-run distribution is found robust in 〈nQ, r, T〉 systems for a large class of demand processes. It is also shown that in 〈R, r, T〉 systems the subsequences of cyclic non-stationary inventory position process associated with cyclic behavior of demand patterns converge in distribution.     

The two-group heuristic to solve the multi-product,economic lot sizing and scheduling problem in flow shops

This paper presents a new and efficient heuristic to solve the multi-product, economic lot sizing and scheduling problem in flow shops. The problem addressed is that of making sequencing, lot sizing and scheduling decisions for a number of products so as to minimize the sum of setup costs, work-in-process inventory holding costs and final-products inventory holding costs while a given demand is fulfilled without backlogging. The proposed heuristic, called the two-group method (TG), assumes that the cycle time of each product is an integer multiple of a basic period and restricts these multiples to take either the value 1 or K where K is a positive integer. The products to be produced once each K basic period are then partitioned into K sub-groups and each sub-group is assigned to one and only one of the K basic periods of the global cycle. This method first determines a value for K and a feasible partition. Then, a production sequence is determined for each sub-group of products and a non-linear program is solved to determine lot sizes and a feasible schedule. We also show how to adapt our method to the case of batch streaming (transportation of sub-batches from one machine to the next). To evaluate its performance, the TG method was compared to both the common cycle method and a reinforced version of El-Najdawi’s job-splitting heuristic. Numerical results show that the TG method outperforms both of these methods.     

The anticipative concept in warehouse optimization using simulation in an uncertain environment

The modern business environment is highly unpredictable. An anticipation approach in a real case study is presented to cope with such instability and minimize the total inventory cost without stock-outs occurring and inventory capacity being exceeded. The anticipation concept is performed using simulation models supported by inventory control algorithms on a selected sample of representative items. The inventory control algorithms include Silver–Meal, Part period balancing, Least-unit cost, and Fuzzy inventory control algorithm based on fuzzy stock-outs, highest inventory level and total cost. Transportation cost is explicitly defined as a discrete function of shipment size. The algorithms are tested on historic data. Simulation results are presented and the risk of accepting them as reliable is discussed. The process of simulation model implementation is briefly discussed to further validate the model and train order managers to use the simulation model in their order placement process.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Supply chain inventory optimization with two customer classes in discrete time

In this paper we consider a single-item inventory system where two demand classes with different service requirements are satisfied from a common inventory. A critical level, reorder point, order quantity or (s, q, k) policy is in use. The time axis is divided into discrete time units, which is a common characteristic of many real-life supply-chain processes. The inventory process within the lead time of a replenishment order is modelled as a sequence of (1) an ordinary renewal process and (2) two alternating renewal processes. Approximations are developed for the demand class-specific fill rates and the probability distribution of the waiting time of low priority customer orders. This waiting time distribution is used for the inventory allocation in a two-stage supply chain.     

The capital cost of holding inventory with stochastically mean-reverting purchase price

Most models of inventory control assume that the per unit purchase price is constant. The capital cost of holding inventory can then be taken into account by adding a fixed interest rate, r, times the purchase price, C, to the out-of pocket holding cost. However, it is not uncommon that the purchase price varies over time. How the capital cost then should be calculated is the focus of the present paper. The paper studies the common single-item inventory model with a fixed set-up cost and assumes that the stochastic purchase price follows the mean-reverting Ornstein–Uhlenbeck process. Methods for computing an adjusted interest rate, r, are suggested along with modifications of well-known heuristics and formulas for lot-sizing. Simulation tests, where the optimal policy has been compared to policies obtained using modified versions of the Silver–Meal method, the Part Period algorithm and the EOQ formula, suggest that r should be estimated as the sum of the unadjusted interest rate and the average expected purchase price decrease, measured over a period between 1/3 and 2/3 of the length of the order cycle.     

Economic lot sizing problem with inventory bounds

This paper studies a economic lot sizing (ELS) problem with both upper and lower inventory bounds. Bounded ELS models address inventory control problems with time-varying inventory capacity and safety stock constraints. An O(n²) algorithm is found by using net cumulative demand (NCD) to measure the amount of replenishment requested to fulfill the cumulative demand till the end of the planning horizon. An O(n) algorithm is found for the special case, the bounded ELS problem with non-increasing marginal production cost.     

Inventory of damagable items with variable replenishment rate,stock-dependent demand and some units in hand

Items made of glass, ceramics, etc., break/get damaged during the storage due to the accumulated stress of heaped stock. For the first time, a deterministic inventory model of such a damagable item is developed with variable replenishment when both demand and damage rates are stock-dependent in polynomial form. Here replenishment rate for the first cycle is partly instantaneous and partly varies with demand. For the next cycle, the variable replenishment is augmented when the inventory level falls to Q₀, the instantaneous replenishment amount for the first cycle. After this, the cycle repeats itself. The amount, Q₀ is also here varied and the optimum Q₀ and Q (inventory level) are evaluated following the profit maximization principle in integral form. The model is illustrated numerically and sensitivity analyses are presented.     

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.     

A deteriorating multi-item inventory model with fuzzy costs and resources based on two different defuzzification techniques

Normally inventory models of deteriorating items, such as food products, vegetables, etc. involve imprecise parameters, like imprecise inventory costs, fuzzy storage area, fuzzy budget allocation, etc. In this paper, we aim to provide two defuzzification techniques for two fuzzy inventory models using (i) extension principle and duality theory of non-linear programming and (ii) interval arithmetic. On the basis of Zadeh’s extension principle, two non-linear programs parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimum average cost at α-level, through which the membership function of the objective function is constructed. In interval arithmetic technique the interval objective function has been transformed into an equivalent deterministic multi-objective problem defined by the left and right limits of the interval. This formulation corresponds to the possibility level, α = 0.5. Finally, the multi-objective problem is solved by a multi-objective genetic algorithm (MOGA). The model has been illustrated through a numerical example and solved for different values of possibility level, α through extension principle and for α = 0.5 via MOGA. As a particular case, the results have been obtained for the inventory model without deterioration. Results from two methods for α = 0.5 are compared.     

PRODUCTION INVENTORY SYSTEM WITH RANDOM SUPPLY INTERRUPTIONS STATUE AND RANDOM LEAD TIMES

This article analyzes a continuous-review inventory system with random supply interruptions and random lead time which may be interrupted by a random number of supplier’s OFF periods. The inventory with constant demand rate is managed by a (r; q1, q2, ··· , qm) policy and supplies from an unreliable sole supplier. By renewal theory and matrix Geometric method, the long-run average cost function is obtained and some important properties of the function are proved. Furthermore, performance of the inventory is derived.     

On the stochastic uncapacitated dynamic single-item lotsizing problem with service level constraints

In this paper, we consider the uncapacitated single-item dynamic lotsizing problem with stochastic period demands and backordering. We present a model formulation that minimizes the setup and holding costs with respect to a constraint on the probability that the inventory at the end of any period does not become negative (α service level) and, alternatively, to a fill rate constraint (β service level). In contrast to earlier model formulations which consider the cycle α service level (α_c) and which approximate the on hand inventory by the net inventory, we include the exact on hand inventory into the model formulation. Therefore, the models are also applicable in situations with very low service levels.