M/M/1 Queueing systems with inventory

We derive stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system. All authors were supported by DAAD/KBN grant number D/02/32206.     

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.     

Integrated capacity and inventory management with capacity acquisition lead times

We model a make-to-stock production system that utilizes permanent and contingent capacity to meet non-stationary stochastic demand, where a constant lead time is associated with the acquisition of contingent capacity. We determine the structure of the optimal solution concerning both the operational decisions of integrated inventory and flexible capacity management, and the tactical decision of determining the optimal permanent capacity level. Furthermore, we show that the inventory (either before or after production), the pipeline contingent capacity, the contingent capacity to be ordered, and the permanent capacity are economic substitutes. We also show that the stochastic demand variable and the optimal contingent capacity acquisition decisions are economic complements. Finally, we perform numerical experiments to evaluate the value of utilizing contingent capacity and to study the effects of capacity acquisition lead time, providing useful managerial insights.     

Dynamic inventory rationing strategies for inventory systems with two demand classes,Poisson demand and backordering

We study inventory systems with two demand classes (critical and non-critical), Poisson demand and backordering. We analyze dynamic rationing strategies where the number of items reserved for critical demand depends on the remaining time until the next order arrives. Different from results in the literature, we do not discretize demand but derive a set of formulae that determine the optimal rationing level for any possible value of the remaining time. Moreover, we show that the cost parameters can be captured in a single relevant dimension, which allows us to present the optimal rationing levels in charts and lookup tables that are easy to implement. Numerical examples illustrate that the optimal dynamic rationing strategy outperforms all static strategies with fixed rationing levels.     

A simple policy for multiple queues with size-independent service times

We consider a service system with two Poisson arrival queues. A server chooses which queue to serve at each moment. Once a queue is served, all the customers will be served within a fixed amount of time. This model is useful in studying airport shuttling or certain online computing systems. We propose a simple yet optimal state-independent policy for this problem which is not only easy to implement, but also performs very well.     

Service constrained inventory models with random lifetimes and lead times

We consider continuous review inventory models with Poisson demands, exponentially distributed lifetimes and replenishment order processing times. The model is unique in that complete backorders, complete lost sales and partial backorders are all addressed. We derive exact expressions of the key operating characteristics. Based on these performance measures, we optimise the system parameters subject to fill rate and waiting time constraints. Numerical examples are also provided to validate our models and to generate useful managerial insights.     

Optimal dynamic pricing and inventory control with stock deterioration and partial backordering

This paper studies the optimal dynamic pricing and inventory control policies in a periodic-review inventory system with fixed ordering cost and additive demand. The inventory may deteriorate over time and the unmet demand may be partially backlogged. We identify two sufficient conditions under which (s,S,p) policies are optimal.     

An algorithm for repairable item inventory system with depot spares and general repair time distribution

We consider the problem of determining the initial spare inventory level for a multi-echelon repairable item inventory system. We extend the previous results to the system, which has an inventory at the central depot as well as at bases and with a general repair time distribution. We propose an algorithm which finds spare inventory level to minimize the total expected cost and simultaneously to satisfy a specified minimum service rate. Extensive computational experiments show that the algorithm is accurate and efficient.     

Calculating safety stocks for assembly systems with random component procurement lead times: A branch and bound algorithm

In this paper, a discrete single-level multi-component inventory control model for assembly systems with random component procurement lead times is considered. The economic order quantity (EOQ) policy is used for a type of finished product. The requirements of the components are constant and cyclic (periodic), and their values per period are deduced from the EOQ for the finished product. The paper focuses on the components safety stock calculation. The objective is to minimise the average holding cost of the components while keeping the desired service level for the finished product. For this, an upper bound, two lower bounds, two dominance properties and an efficient branch and bound algorithm are suggested. Several tests are executed and conclusions are drawn. The proposed model provides a substantial saving for assembly systems with a large number and unreliable delivery of components as in semi-conductor and automotive industries.     

Note on the fill rate of single-stage general periodic review inventory systems

For base-stock policies, Zhang and Zhang [J. Zhang, J. Zhang, Fill rate of single-stage general periodic review inventory systems, Operations Research Letters 35 (2007) 503-509] derive the fill rate, defined as the long-run average fraction of demand satisfied immediately. We derive the same expression for the fill rate defined as the ratio of expected demand satisfied immediately to expected demand, and generalize to (R,Q) policies.     

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.     

Optimal policies for inventory systems with discretionary sales, random yield and lost sales

We determine replenishment and sales decisions jointly for an inventory system with random demand, lost sales and random yield. Demands in consecutive periods are independent random variables and their distributions are known. We incorporate discretionary sales, when inventory may be set aside to satisfy future demand even if some present demand may be lost. Our objective is to minimize the total discounted cost over the problem horizon by choosing an optimal replenishment and discretionary sales policy. We obtain the structure of the optimal replenishment and discretionary sales policy and show that the optimal policy for finite horizon problem converges to that of the infinite horizon problem. Moreover, we compare the optimal policy under random yield with that under certain yield, and show that the optimal order quantity （sales quantity） under random yield is more （less） than that under certain yield.     

A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times

This paper considers a number-dependent replacement policy for a system with two failure types that is replaced at the nth type I (minor) failure or the first type II (catastrophic) failure, whichever occurs first. Repair or replacement times are instantaneous but spare/replacement unit delivery lead times are random. Type I failures are repaired at zero cost since preventive maintenance is performed continuously. Type II failures, however, require costly system replacement. A model is developed for the average cost per unit time based on the stochastic behavior of the system and replacement, storage, and downtime costs. The cost-minimizing policy is derived and discussed. We show that the optimal number of type I failures triggering replacement is unique under certain conditions. A numerical example is presented and a sensitivity analysis is performed.     

Trade-off between capacity invested and inventory needed

A multi-item capacitated make-to-order production system with considerable demand fluctuations is discussed. The relationship between the available capacity and the inventory needed to meet customer requirements with a pre-defined service level is modeled. Furthermore, the total cost for both capacity and inventory is minimized and it is shown that, assuming negligible change-over times, the double of the surplus inventory cost has to be equal to the excess capacity cost to ensure minimum total cost.     

An inventory model with partial backordering and unit backorder cost linearly increasing with the waiting time

As the implementation of JIT practice becomes increasingly popular, each echelon in a supply chain tends to carry fewer inventories, and thus the whole supply chain is made more vulnerable to lost sales and/or backorders. The purpose of this paper is to recast the inventory model to be more relevant to current situations, where the penalty cost for a shortage occurrence at a downstream stage in a supply chain is continually transmitted to the upstream stages. The supplier, in this case, at the upstream of the supply chain is responsible for all the downstream shortages due to the chain reaction of its backlog. The current paper proposes a model in which the backorder cost per unit time is a linearly increasing function of shortage time, and it claims that the optimal policy for the supplier is setting the optimal shortage time per inventory cycle to minimize its total relevant cost in a JIT environment.     

A multi-item mixture inventory model involving random lead time and demand with budget constraint and surprise function

This study deals with a multi-item mixture inventory model in which both demand and lead time are random. A budget constraint is also added to this model. The optimization problem with budget constraint is then transformed into a multi-objective optimization problem with the help of fuzzy chance-constrained programming technique and surprise function. In our studies, we relax the assumption about the demand, lead time and demand during lead time that follows a known distribution and then apply the minimax distribution free procedure to solve the problem. We develop an algorithm procedure to find the optimal order quantity and optimal value of the safety factor. Finally, the model is illustrated by a numerical example.     

Inventory sharing in integrated network design and inventory optimization with low-demand parts

Service Parts Logistics (SPL) problems induce strong interaction between network design and inventory stocking due to high costs and low demands of parts and response time based service requirements. These pressures motivate the inventory sharing practice among stocking facilities. We incorporate inventory sharing effects within a simplified version of the integrated SPL problem, capturing the sharing fill rates in 2-facility inventory sharing pools. The problem decides which facilities in which pools should be stocked and how the demand should be allocated to stocked facilities, given full inventory sharing between the facilities within each pool so as to minimize the total facility, inventory and transportation costs subject to a time-based service level constraint. Our analysis for the single pool problem leads us to model this otherwise non-linear integer optimization problem as a modified version of the binary knapsack problem. Our numerical results show that a greedy heuristic for a network of 100 facilities is on average within 0.12% of the optimal solution. Furthermore, we observe that a greater degree of sharing occurs when a large amount of customer demands are located in the area overlapping the time windows of both facilities in 2-facility pools.     

Joint management of capacity and inventory in make-to-stock production systems with multi-class demand

We evaluate the benefits of coordinating capacity and inventory decisions in a make-to-stock production environment. We consider a firm that faces multi-class demand and has additional capacity options that are temporary and randomly available. We formulate the model as a Markov decision process (MDP) and prove that a solution to the optimal joint control problem exists. For several special cases we characterize the structure of the optimal policy. For the general case, however, we show that the optimal policy is state-dependent, and in many instances non-monotone and difficult to implement. Therefore, we consider three pragmatic heuristic policies and assess their performance. We show that the majority of the savings originate from the ability to dynamically adjust capacity, and that a simple heuristic that can adjust production capacity (based on workload fluctuation) but uses a static production/rationing policy can result in significant savings.     

Uniform distribution of inventory positions in two-echelon periodic review systems with batch-ordering policies and interdependent demands

We consider a two-level inventory system in which there are one supplier and multiple retailers. The retailers face stochastic, interdependent customer demands. Each location employs a periodic-review (R,nQ), or lot-size reorder point, inventory policy. We show that each location's inventory positions are stationary and the stationary distribution is uniform and independent of any other's.