Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations and Perturbation Analysis

We consider a model of a supply chain consisting of n production facilities in tandem and producing a single product class. External demand is met from the finished goods inventory maintained in front of the most downstream facility (stage 1); unsatisfied demand is backlogged. We adopt a base-stock production policy at each stage of the supply chain, according to which the facility at stage i produces if inventory falls below a certain level w _i and idles otherwise. We seek to optimize the hedging vector w=(w ₁,...,w _n ) to minimize expected inventory costs at all stages subject to maintaining the stockout probability at stage 1 below a prescribed level (service level constraint). We make rather general modeling assumptions on demand and production processes that include autocorrelated stochastic processes. We solve this stochastic optimization problem by combining analytical (large deviations) and sample path-based (perturbation analysis) techniques. We demonstrate that there is a natural synergy between these two approaches.     

Simple approximations for the reorder point in periodic and continuous review (s, S) inventory systems with service level constraints

In this paper we consider a general class of (s, S) inventory systems including periodic review and continuous review systems. We allow for stochastic lead times for replenishment orders provided that the probability of orders crossing in time is negligible for the relevant (s, S) control rules. In accordance with common practice we emphasize on service level constraints rather than assuming given stockout costs. In particular we consider the service measure requiring that a specified fraction of the demand is met directly from stock on hand. The purpose of this paper is to present practically useful approximations for the recorder point s such that the required service level is achieved. By a simple and direct approach, a unifying treatment of the general class of (s, S) inventory systems considered is given. We obtain for the first time tractable approximations for the continuous review (s, S) inventory system with undershoots of the reorder point. Also, we discuss 2-moments approximations obtained by fitting normal respectively gamma distributions to the empirical demand distributions. Extensive numerical experience with the approximations is reported, including results about the sensitivity of the reorder point to the higher moments of the demand distributions.     

An inventory control system for products with optional components under service level and budget constraints

Modularization and customization have made enterprises face the multi-item inventory problems and the interactions among those items. A powerful, affordable information technology system can make the continuous review inventory policy more convenient, efficient, and effective. In this study, a (Q, r) model is developed to find the optimal lot size and reorder point for a multi-item inventory with interactions between necessary and optional components. In order to accurately approximate costs, the service cost is introduced and defined in proportion to the service level. In addition, the service cost and purchasing cost are taken simultaneously, and are treated as a budget constraint for executives to consider because the firm’s strategy could influence the choice of service level. The proposed model is formulated as a nonlinear optimization problem, as the service level is nonlinear. Thus, some known procedures are revised to solve this problem and the results are compared with other models. The results show that the revised procedure performs better than the N–R procedure, leading to important insights about inventory control policy.     

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.     

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.     

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.     

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.     

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.     

The impact of data collection on fill rate performance in the (R,s, Q) inventory model

In this paper we consider the determination of the reorder point s in an (R, s, Q) inventory model subject to a fill rate service level constraint. We assume that the underlying demand process is a compound renewal process. We then derive an approximation method to compute the reorder level such that a target service level is achieved. Restrictions on the input parameters are given, within which this method is applicable. Moreover, we will investigate the effects on the fill rate performance in case the underlying demand process is indeed a compound renewal process, while the demand process is modelled as a discrete-time demand process. That is, the time axis is divided in time units (for example, days) and demands per time unit are independent and identically distributed random variables. It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.     

Covering location problem of emergency service facilities in an uncertain environment

In practical location problems on networks, the response time between any pair of vertices and the demands of vertices are usually indeterminate. This paper employs uncertainty theory to address the location problem of emergency service facilities under uncertainty. We first model the location set covering problem in an uncertain environment, which is called the uncertain location set covering model. Using the inverse uncertainty distribution, the uncertain location set covering model can be transformed into an equivalent deterministic location model. Based on this equivalence relation, the uncertain location set covering model can be solved. Second, the maximal covering location problem is investigated in an uncertain environment. This paper first studies the uncertainty distribution of the covered demand that is associated with the covering constraint confidence level α. In addition, we model the maximal covering location problem in an uncertain environment using different modelling ideas, namely, the (α, β)-maximal covering location model and the α-chance maximal covering location model. It is also proved that the (α, β)-maximal covering location model can be transformed into an equivalent deterministic location model, and then, it can be solved. We also point out that there exists an equivalence relation between the (α, β)-maximal covering location model and the α-chance maximal covering location model, which leads to a method for solving the α-chance maximal covering location model. Finally, the ideas of uncertain models are illustrated by a case study.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.     

m-scheduling-period inventory model for deteriorating items with instantaneous demand

A probabilistic inventory model is developed for deteriorating items for a system that operates exactly for m(m?2) scheduling periods. Deterioration is assumed to be a constant fraction of the on hand inventory; and demand x_n in nth period is assumed to be instantaneous immediately after the nth decision regarding the order-level S_n has been made at the beginning of the nth period (n = 1,2,…,m) the x_n being mutually independent. An example is given to illustrate the model.     

A stochastic production planning model with a dynamic chance constraint

This paper deals with the optimal production planning for a single product over a finite horizon. The holding and production costs are assumed quadratic as in Holt, Modigliani, Muth and Simon (HMMS) [7] model. The cumulative demand is compound Poisson and a chance constraint is included to guarantee that the inventory level is positive with a probability of at least α at each time point. The resulting stochastic optimization problem is transformed into a deterministic optimal control problem with control variable and of the optimal solution is presented. The form of state variable inequality constraints. A discussion the optimal control (production rate) is obtained as follows: if there exists a time t₁ such that t₁?[O, T]where T is the end of the planning period, then (i) produce nothing until t₁ and (ii) produce at a rate equal to the expected demand plus a ‘correction factor’ between t₁ and T. If t₁ is found to be greater than T, then the optimal decision is to produce nothing and always meet the demand from the inventory.     

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.     

An exact solution procedure for multi-item two-echelon spare parts inventory control problem with batch ordering in the central warehouse

We consider a multi-item two-echelon inventory system in which the central warehouse operates under a (Q,R) policy, and the local warehouses implement basestock policy. An exact solution procedure is proposed to find the inventory control policy parameters that minimize the system-wide inventory holding and fixed ordering cost subject to an aggregate mean response time constraint at each facility.     

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].     

A transportation type aggregate production model with bounds on inventory and backordering

We consider a certain T period aggregate production planning model, where the two sources of production are regular and overtime. The model allows for time varying production, holding and backordering costs and includes bounds on inventory and backorders. We show that the problem has a rather interesting network structure and exploit this structure to develop a greedy algorithm to solve the problem. The procedure is easy to implement and has a computational complexity of O(T²). We report computational experience with the greedy procedure and demonstrate its superiority to a well known network simplex code, Gnet, implemented on the classical network formulation of the problem.     

Inventory control for point-of-use locations in hospitals

Most inventory management systems at hospital departments are characterised by lost sales, periodic reviews with short lead times, and limited storage capacity. We develop two types of exact models that deal with all these characteristics. In a capacity model, the service level is maximised subject to a capacity restriction, and in a service model the required capacity is minimised subject to a service level restriction. We also formulate approximation models applicable for any lost-sales inventory system (cost objective, no lead time restrictions etc). For the capacity model, we develop a simple inventory rule to set the reorder levels and order quantities. Numerical results for this inventory rule show an average deviation of 1% from the optimal service levels. We also embed the single-item models in a multi-item system. Furthermore, we compare the performance of fixed order size replenishment policies and (R,?s,?S) policies.     

Stock Rationing in a Continuous Review Two-Echelon Inventory Model

In this paper we consider a 1-warehouse, N-retailer inventory system where demand occurs at all locations. We introduce an inventory model which allows us to set different service levels for retailers and direct customer demand at the warehouse. For each retailer a critical level is defined, such that a retailer replenishment order is delivered from warehouse stock if and only if the stock level exceeds this critical level. It is assumed that retailer replenishment orders, which are not satisfied from warehouse stock, are delivered directly from the outside supplier, instead of being backlogged. We present an analytical upper bound on the total cost of the system, and develop a heuristic method to optimize the policy parameters. Numerical experiments indicate that our technique provides a very close approximation of the exact cost. Also, we show that differentiating among the retailers and direct customer demand can yield significant cost reductions.