Optimal batch ordering policies for assembly systems with guaranteed service

For a supply chain modelled as a multi-echelon inventory system, effective management of its inventory at each stock is critical to reduce inventory costs while assuring a given service level to customers. In our previous work, we used the guaranteed-service approach (GSA) to design optimal echelon batch ordering policies for continuous-review serial systems with Poisson customer demand and fixed order costs. The approach assumes that the final customer demand is bounded and each stock has a guaranteed service time in the sense that the demand of its downstream stock can always be satisfied in the service time. This paper extends this work by considering more general assembly systems. We first derive an analytical expression for the total cost of the system in the long run. The problem of finding optimal echelon batch ordering policies for the system can then be decomposed into two independent sub-problems: order size decision sub-problem and reorder point decision sub-problem. We develop efficient dynamic programming algorithms for the two sub-problems. Numerical experiments on randomly generated instances show the effectiveness of the proposed approach.     

On deficiencies of common ordering policies for multi-level inventory control

We consider two common types of ordering policies for multi-level inventory control: installation stock (R,Q)-policies and echelon stock (R,Q)-policies. The batch quantities are assumed to be given, but each policy is optimized with respect to its reorder point R. We demonstrate that there is no bound for the worst case performance ratio of these policies when applied to distribution inventory systems with a central warehouse and a number of retailers.     

Inventory performance of some supply chain inventory policies under impulse demands

This paper attempts to study the impact of impulsive demand disturbances on the inventory-based performance of some inventory control policies. The supply chain is modelled as a network of autonomous supply chain nodes. The customer places a constant demand except for a brief period of sudden and steep change in demand (called demand impulse). Under this setting, the behaviour of each inventory policy is analysed for inventory performance of each node. It is found that the independent decision-making by each node leads to a bullwhip effect in the supply chain whereby demand information is amplified and distorted. However, under a scenario where the retailer places a constant order irrespective of the end customer demand, the inventory variance was actually found to decrease along the supply chain. The variance of the inventory remained constant along the chain when only the actual demands are transmitted by each node. The results also showed that the inventory policy which is best for one supply chain node is generally less efficient from a supply chain perspective. Moreover, the policy which performs poorly for one node can be most efficient for the supply chain. In a way, our results also provide a case for coordinated inventory management in the supply chain where all members prepare a joint inventory management policy that is beneficial for all the supply chain nodes. The results have significant industrial implications.     

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.     

Effective bullwhip metrics for multi-echelon distribution systems under order batching policies with cyclic demand

A large number of problems in a distribution supply chain require that decisions are made in the presence of the bullwhip effect phenomenon. The impact of the order batching policies on the bullwhip effect is analysed in this paper, when cycle demand on a multi-echelon supply chain operating is considered. While investigating which bullwhip effect metrics are more adequate to measure the bullwhip effect in these type of systems, the optimal reordering plan that minimises the operation costs of the overall system is calculated. A Mixed Integer Linear Programming (MILP) model is developed that takes into account an inventory and distribution system formed by multiple warehouses and retailers with lateral transshipments. The bullwhip effect is measured through four metrics: the echelon average inventory; the echelon inventory variance ratio; the echelon average order; and the echelon order rate variance ratio. As conclusion the inventory metrics suggest that (i) using batching policy reduces instability; (ii) batching may reduce in general order variance if using larger batches and (iii) cycle demand length has no major impact in the bullwhip effect. A motivational example and a real word case study are used and tested.     

Effect of modeling fixed cost in a serial inventory system with periodic review

We study a two-echelon serial inventory system with stochastic demand. We assume that fixed ordering costs are charged only when an order initiates a non-zero shipment. The system is centrally controlled and ordering decisions are based on echelon base stock policies. The review period of the upper echelon is an integer multiple of the review period of the lower echelon. We derive an exact analytical expression for the objective function. From this expression, we determine optimal base stock levels and review periods. Through a numerical study we show that there may be several combinations of optimal review periods and that under high fixed ordering costs both stockpoints have the same order frequency. In addition, we identify parameter settings under which the system behaves like a PUSH-system, where the upstream stockpoint never holds any stock. Generally, in literature fixed ordering costs are charged at every review moment, even if no shipment results due to zero upstream stock. We test the impact of this simplifying assumption and illustrate when it is justified.     

Decentralized inventory control in a two-stage capacitated supply chain

This paper investigates a two-stage supply chain consisting of a capacitated supplier and a retailer that faces a stationary random demand. Both the supplier and the retailer employ base stock policies for inventory replenishment. All unsatisfied demand is backlogged and the customer backorder cost is shared between the supplier and the retailer. We investigate the determination of decentralized inventory decisions when the two parties optimize their individual inventory-related costs in a noncooperative manner. We explicitly characterize the Nash equilibrium inventory strategies and identify the causes of inefficiency in the decentralized operation. We then study a set of simple linear contracts to see whether these inefficiencies can be overcome. Finally, we investigate Stackelberg games where one of the parties is assumed to be dominating.     

Analysis of a deterministic demand production/inventory system under nonstationary supply uncertainty

In this article we investigate a periodic review inventory model under deterministic dynamic demand and supply unavailability. In a given period, supply is either available or completely unavailable with given probabilities. Supply unavailability probabilities are nonstationary over time. We show the optimality of an order-up-to level policy, and obtain a newsboy-like formula that determines the optimal order-up-to levels. Our formula would provide guidence as to the appropriate amount of inventory to stock in the face of uncertainties in the supply process.     

Optimal policies of a two-echelon serial inventory system with general limited capacities

We study optimal policies of capacitated two-echelon serial inventory systems under periodic review. For a system with smaller downstream capacity, we fully characterise the optimal policy as a further modified echelon base stock policy using an intuitive backward induction. The key lies in the magnitude relation between the initial upstream stock level and the downstream capacity. For a system with smaller upstream capacity, we demonstrate that the optimal policy is of a more complex structure where there can be at most four/five target levels up to which the upstream/downstream echelon tries to produce/order. The numbers of levels and their values depend on the length of remaining horizons and the amount of initial upstream inventories. We also specify these potential target levels and then suggest a way to simplify the search of optimal solutions.     

Service differentiation in a single-period inventory model with numerous customer classes

We study critical-level inventory-management policies as means to provide differentiated (\(\alpha \) and \(\beta \)) service levels to more than two classes of customers. First, we derive closed-form expressions for the service levels of a single-period critical-level policy with an arbitrary number of customer classes (with Poisson demand). Based on the service-level expressions, we derive additional structural insights and provide an efficient algorithm with which to compute the essential system parameters, that is, the minimum required starting inventory and the associated critical levels. Based on these results, we conduct numerical experiments and develop structural insights into the system’s behavior.     

Base-stock control for single-product tandem make-to-stock systems

In this paper, we consider a multiple-stage tandem production/inventory system producing a single product. Processing time at each stage is assumed to have a general stationary processing time distribution. The cost of holding work-in-process (WIP) inventory is different at each stage. Therefore, decisions on when to release work to the system as well as when to transfer WIP from one stage to another need to be made. We formulate this problem of release/production control as a Markov decision process. However, the optimal policy is rather complex, making its implementation impracticable in practice. We therefore investigate the performance of simple base stock policies. Our approach aggregates several stages into one and uses a simple approximation to compute 'approximately optimal' base stock levels. We present the results of a simulation study that tests the performance of our approximation in estimating the best base stock levels, and the performance of base stock policies as compared with the optimal policy.     

Impact of the pull and push‐pull policies on the performance of a three‐stage supply chain

We investigate the performance of two common operating policies (i.e., pull and push‐pull) for a make‐to‐stock product in an un‐capacitated, three‐stage supply chain. The pull policy operates based on periodic orders received from the immediate downstream facility. However, in the push‐pull policy, while processes upstream of the order decoupling point are managed by the push policy, the downstream processes are managed by the pull policy. Simulation experiments are conducted to examine the impact of each operating policy under a variety of experimental conditions, characterised by demand uncertainty and lead‐time variability. Our results indicate that the relative advantage of the two policies is dependent on the type of uncertainty, the level of uncertainty, the inventory control policy and the performance measures of interest. More specifically, while the push‐pull policy results in lower inventory, the pull policy yields a better fill rate. This is in contrast to the notion that the pull policy generally results in superior inventory performance. Our findings suggest that firms should carefully consider the level of uncertainty, the inventory control policy and the performance measures of interest when determining the operating policy.     

The impact of quality considerations on material flow in two-stage inventory systems

Supply chain management literature calls for coordination between the different members of the chain. Inventory models achieve this coordination along a supply chain by making the lot size at an upstream entity an integer multiplier of the lot size at the adjacent downstream entity. Such models assume that all components produced are of acceptable quality and may cause suppliers to produce larger quantities than what is optimal. In this paper, we formulate and solve two-stage supply chain inventory models in which the proportion of defective products increases with increased production lot sizes. We show that quality considerations can lead to significant reduction in production lot sizes. In addition, the models show that most benefits to the supply chain are attained from the suppliers producing on a just-in-time basis rather than delivering to their customers just-in-time. We derive closed-form expressions for the optimal lot sizes for a two-stage supply chain under deterministic and then stochastic demand and illustrate the models with numerical examples.     

A stochastic inventory model for an immediate liquidation and price-promotion decision under price-dependent demand

We consider a periodic review stochastic inventory system where the current on-hand inventory exceeds the maximum supply needs in the future. Consequently, one must make an immediate inventory liquidation decision on the liquidation quantity and promotional price with the goal of maximising the overall profit where the demand during the liquidation period (DDLP) is a random variable whose distribution depends on the promotional price. We develop a price-dependent DDLP model and an inventory model for optimising the liquidation quantity and unit promotional price. The model is applicable for general distributions of the DDLP and regular demand (i.e. demand during the future periods following the promotion period). We also investigate four special cases where the DDLP and regular demand are assumed to be either exponential or uniform random variables. The two models that assume the exponential distribution for regular demand can be examined analytically and simplified using the mathematical properties we derive. The additional two models that assume the uniform distribution for regular demand do not have closed-form expressions but can be solved numerically. Some numerical examples are presented for further elaboration of the models and to demonstrate their practical use.     

Stockout propagation and amplification in supply chain inventory systems

We investigate the existence and magnitude of stockout propagation and stockout amplification in the context of supply chain inventory systems. Stockout amplification is a stage-to-stage increase in overall stockout rates. Stockout propagation is the tendency for stockout at one node to instigate a stockout at a neighbouring node and is conceptually related to the idea of cascading failures in physical systems, such as electrical power grids. We study these concepts in both upstream (‘supply side’) and downstream (‘demand side’) directions in the context of normal operating conditions for an adaptive R, S (periodic, order-up-to) inventory policy. We build a simulation model of a 5-stage serial supply chain that experiences normally distributed customer demands and gamma distributed lead times. We find that stockout propagation exists, but contrary to conventional wisdom, it occurs in the upstream direction. There is little indication that stockout propagation is occurring to any significant degree in the downstream direction. We also find stockout amplification occurring in the upstream direction in scenarios where more aggressively adaptive inventory parameter updating is performed. We discuss implications of this work in the areas of supply chain inventory modelling, ordering decisions, safety stock determination, and the use of adaptive inventory policies.     

The power of coordinated decisions for short-life-cycle products in a manufacturing and distribution supply chain

This paper explores ways of conceptualizing and studying the power of coordination and strategic alliances within a manufacturing and distribution supply chain consisting of one manufacturer and one distributor. The supply chain operates to meet price-sensitive random demand of a product with a short product life cycle. We derive closed-form multi-attribute measures of performance for the supply chain in the presence of and in the absence of coordination of pricing and production/ordering decisions. The optimal coordinated pricing and production/ordering policies, as well as each party's optimal policies in the absence of coordination, are developed. Our results yield insights into basic managerial issues for this type of manufacturing and distribution supply chains.     

Comparative modelling of multi-stage production-inventory control policies with lot sizing

We propose a unified modelling framework, based on a queuing network representation, for describing, comparing and contrasting classical and new multi-stage production-inventory control policies with lot sizing. The classical policies that we consider are installation stock reorder point policies, echelon stock reorder point policies and classical kanban policies, which we refer to as installation kanban policies. The new policies that we propose are echelon kanban policies and hybrid policies, which combine reorder point and kanban policies. The combination of reorder point and kanban policies can be done in a synchronized or an independent way, leading to synchronized and independent hybrid policies, respectively.     

Models for integrated production-inventory systems: steady state and cost analysis

We consider a two-echelon production-inventory system with a central supplier connected to production systems (servers) at several locations, each with a local inventory. Demand of customers arrives at each production system according to a Poisson process and is lost if the local inventory is depleted. To satisfy a customer’s demand, a server at the production system takes exactly one unit of raw material from the associated local inventory. The central supplier manufactures raw material to replenish the local inventories, which are controlled by a continuous review base stock policy. We derive stationary distributions of joint queue length and inventory processes in explicit product form. After performing a cost analysis, we find out that the global search for the vector of optimal base stock levels can be reduced to a set of independent optimisation problems. The explicit form of the stationary distribution enables us to get additional structural insights, e.g. about monotonicity properties and stability conditions. Obtaining the product form relies on some simplifying assumptions. The results are therefore compared with simulations of a more realistic system, which supports to use it as approximation.     

Base-stock control for single-product tandem make-to-stock systems

In this paper, we consider a multiple-stage tandem production/inventory system producing a single product. Processing time at each stage is assumed to have a general stationary processing time distribution. The cost of holding work-in-process (WIP) inventory is different at each stage. Therefore, decisions on when to release work to the system as well as when to transfer WIP from one stage to another need to be made. We formulate this problem of release/production control as a Markov decision process. However, the optimal policy is rather complex, making its implementation impracticable in practice. We therefore investigate the performance of simple base stock policies. Our approach aggregates several stages into one and uses a simple approximation to compute approximately optimal base stock levels. We present the results of a simulation study that tests the performance of our approximation in estimating the best base stock levels, and the performance of base stock policies as compared with the optimal policy.     

An improved ordering policy for continuous review inventory systems with arbitrary inter-demand time distributions

In this paper, we revisit the optimality of (s,S) policies in continuous review inventory models where demand forms a renewal process. We explain why when orders are placed at demand epochs, (s,S) policies are not optimal in general, and propose a simple but practical ordering policy by introducing a delay in order placement as a policy parameter. Under our proposed policy, the operating characteristics of such systems can be evaluated using the existing results in the literature. In order to demonstrate the effectiveness of our policy, we restrict our analysis to a special class of inventory systems where demand follows a counting process, fixed ordering costs are negligible and leadtimes are constant. We derive expressions for the operating characteristics of such systems under our policy and present insights on the behavior of the policy parameters which minimize the average total cost rate under our policy. Using these results, we develop an efficient heuristic for computing the policy parameters. Finally, we investigate the effectiveness of our policy in comparison to policies which place orders only at demand epochs. The results of our numerical experiment indicate that our policy can result in a significant savings.