The effects of transportation lead-time and safety stock in international co-operative knockdown production systems

Abstract

Abstract. In international co-operative knockdown production systems (ICKDPS), much longer transportation lead-time than production lead-time to procure component parts is required, because the parts are generally containerized by ship. In this paper, in order to determine an economical buffer stock level for a more reliable ICKDPS, a pull-type ordering system which considers the backorder for compensating for shortage of component parts is developed for the ICKDPS as a three-stage production, inventory and transportation system with the concept of 'just-in-time’ production systems. In addition, the effects of transportation lead-time and buffer stock level on the quantity of backlogs and the probability of requirement are clarified by simulation.     

具有多元马氏需求的多产品多阶段库存优化模型

本文研究一类新的多产品库存控制策略,即具有多元马氏需求特征的多产品多阶段的订货点订货量(Q, R, SS)策略,该策略考虑市场需求在不同产品之间具有多元马氏转移特征,并考虑缺货因素设置安全库存。论文首先建立了多产品多阶段的多元马氏需求预测模型,并通过该模型确定了各种产品需求之间的关系。同时,在该模型的理论基础上,提出了多产品多阶段的总期望成本模型及其最优(Q, R, SS)策略,进而结合算例给出模型的最优策略的数值解。     

Optimal Ordering Policies for Inventory Problems With Dynamic Information Delays

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. We introduce dynamic information delays as a Markov process into the standard multiperiod stochastic inventory problem with backorders. We develop the concept of a reference inventory position. We show that this position along with the magnitude of the latest observed delay and the age of this observation are sufficient statistics for finding the optimal order quantities. Furthermore, we establish that the optimal ordering policy is of state‐dependent base‐stock type with respect to the reference inventory position (or state‐dependent (s, S) type if there is a fixed ordering cost). The optimal base stock and (s, S) levels depend on the magnitude of the latest observed delay and the age of this observation. Finally, we study the sensitivity of the optimal base stock and the optimal cost with respect to the sufficient statistics.     

Inventory control with seasonality of lead times

The practical challenges posed by the seasonality of lead times have largely been ignored within the inventory control literature. The length of the seasons, as well as the length of the lead times during a season, may demonstrate cyclical patterns over time. This study examines whether inventory control policies that anticipate seasonal lead-time patterns can reduce costs. We design a framework for characterizing different seasonal lead-time inventory problems. Subsequently, we examine the effect of deterministic and stochastic seasonal lead times within periodic review inventory control systems. We conduct a base case analysis of a deterministic system, enabling two established and alternating lead-time lengths that remain valid through known intervals. We identify essential building blocks for developing solutions to seasonal lead-time problems. Lastly, we perform numerical experiments to evaluate the cost benefits of implementing an inventory control policy that incorporates seasonal lead-time lengths. The findings of the study indicate the potential for cost improvements. By incorporating seasonality in length of seasons and length of lead times within the season into the control models, inventory controllers can make more informed decisions when ordering their raw materials. They need smaller buffers against lead-time variations due to the cyclical nature of seasonality. Reductions in costs in our experiments range on average between 18.9 and 26.4% (depending on safety time and the probability of the occurrence of stock out). Therefore, inventory control methods that incorporate seasonality instead of applying large safety stock or safety time buffers can lead to substantial cost reductions.     

Optimal Inventory Control with Retail Pre‐Packs

A pre‐pack is a collection of items used in retail distribution. By grouping multiple units of one or more stock keeping units (SKU), distribution and handling costs can be reduced; however, ordering flexibility at the retail outlet is limited. This paper studies an inventory system at a retail level where both pre‐packs and individual items (at additional handling cost) can be ordered. For a single‐SKU, single‐period problem, we show that the optimal policy is to order into a “band” with as few individual units as possible. For the multi‐period problem with modular demand, the band policy is still optimal, and the steady‐state distribution of the target inventory position possesses a semi‐uniform structure, which greatly facilitates the computation of optimal policies and approximations under general demand. For the multi‐SKU case, the optimal policy has a generalized band structure. Our numerical results show that pre‐pack use is beneficial when facing stable and complementary demands, and substantial handling savings at the distribution center. The cost premium of using simple policies, such as strict base‐stock and batch‐ordering (pre‐packs only), can be substantial for medium parameter ranges.     

An echelon inventory-based single-stage cost function for a two-station tandem system

We propose an alternative cost-accounting function for inventory control problems on a make-to-stock setting. Our proposal is based on observing that the traditional holding and backlog parameters introduce some odd short term distortions on the inventory state space. Our single-stage cost function accounts for echelon inventories and possesses a pair of cost parameters for each echelon inventory variable, depending on whether it is positive or negative. With the modified cost-accounting function, we study a twostation tandem system producing a single product, and investigate how it compares with the performances obtained with the usual single-stage cost function. The results available so far show that the optimal policies approach a multi-echelon base stock structure for each machine. Also, the service levels achieved are better under the modified function without increasing the levels of finished goods inventory.     

An arborescent inventory model in a supply chain system

This study develops an arborescent (tree-like) inventory model with a constant demand rate. By considering the integration of the producer, distributors and retailers, a mathematical model and an economic ordering policy are developed. It can be shown that the integrated approach results in an impressive cost-reduction compared with an independent decision by the partial party. A numerical example of an arborescent inventory system is attached to demonstrate the economic ordering policy with the help of mathematical software - Maple V.     

POWER APPROXIMATIONS FOR A TWO-ECHELON INVENTORY SYSTEM USING SERVICE LEVELS

We consider a two-echelon inventory system with one warehouse and several stores. The warehouse as well as the stores are controlled by periodic review (s, S) inventory policies. We study the interrelationship between the safety stocks at the warehouse and the stores. Stockouts at the warehouse will result in supply delays to the stores and cause the lead time to be stochastic. The stores may react by increasing their safety stock. However, there is a trade-off between the safety stock at the warehouse and the safety stock at the stores. We use a service level at the warehouse to quantify the effect of warehouse stockouts on the lead time to the stores. The service level at the warehouse is considered a decision variable to find the best compromise between the various safety stocks by minimizing the overall costs. Using power approximations for the (s, S) policies, we provide an iterative procedure for adjusting the lead time distribution to the stores; this can result in substantial savings, but it doesn't guarantee the overall optimality. Numerical studies are provided to test the accuracy of approximations. The effects of the different system parameters on the inventory policy give general guidelines for use of the policies.     

Studies of optimal ordering policies for perishable commodities

Abstract

Problems associated with perishable or outdating products arise in many areas of inventory and production man agement. Many commodities exist whose value does not remain constant over time during transportation, holding stock, etc. Significant work has been done to describe optimal ordering policies for items which have a fixed lifetime. In this paper, per ishable inventory models subject to stochastic procurement leadtinie both on LIFO (Last In First Out) and FIFO (First In First Out) issuing policies are considered with cither zero or 1 unit leadtinie. First, assumptions and notation used throughout the paper are given. FIFO and LIFO models are then con structed, and the existence of the optimal ordering policies and theirpropertiesare clarified. Finally, significantresults are illustrated for the relationship among inventory on hand, delay probability of procurement and optimal ordering policies.     

Experimental Results Indicating Lattice‐Dependent Policies May Be Optimal for General Assemble‐To‐Order Systems

We consider an assemble‐to‐order (ATO) system with multiple products, multiple components which may be demanded in different quantities by different products, possible batch ordering of components, random lead times, and lost sales. We model the system as an infinite‐horizon Markov decision process under the average cost criterion. A control policy specifies when a batch of components should be produced, and whether an arriving demand for each product should be satisfied. Previous work has shown that a lattice‐dependent base‐stock and lattice‐dependent rationing (LBLR) policy is an optimal stationary policy for a special case of the ATO model presented here (the generalized M‐system). In this study, we conduct numerical experiments to evaluate the use of an LBLR policy for our general ATO model as a heuristic, comparing it to two other heuristics from the literature: a state‐dependent base‐stock and state‐dependent rationing (SBSR) policy, and a fixed base‐stock and fixed rationing (FBFR) policy. Remarkably, LBLR yields the globally optimal cost in each of more than 22,500 instances of the general problem, outperforming SBSR and FBFR with respect to both objective value (by up to 2.6% and 4.8%, respectively) and computation time (by up to three orders and one order of magnitude, respectively) in 350 of these instances (those on which we compare the heuristics). LBLR and SBSR perform significantly better than FBFR when replenishment batch sizes imperfectly match the component requirements of the most valuable or most highly demanded product. In addition, LBLR substantially outperforms SBSR if it is crucial to hold a significant amount of inventory that must be rationed.     

Inventory‐Based Dynamic Pricing with Costly Price Adjustment

We study an average‐cost stochastic inventory control problem in which the firm can replenish inventory and adjust the price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size—An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time‐average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory‐based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.     

Evaluation of safety stock methods in multilevel material requirements planning (MRP) systems

This paper evaluates alternative methods of establishing the safety stock level taking into consideration of historical measures of forecasting accuracy and the needs for master production scheduling and material requirement planning under a rolling time horizon. A computer model is used to simulate the forecasting, master production scheduling and material planning activities in a company that produces to stock and the production activities are managed by multilevel MRP systems. The simulation output is analysed to evaluate the impact of safety stock methods on MRP system performance. The result of the study shows that using safety stock can help to reduce total cost, schedule instability and improve service level in the MRP systems. Guidelines are developed to help managers select methods to determine safety stock in MRP system operations.     

Optimal Ordering Policies for Stochastic Inventory Problems with Observed Information Delays

Information delays exist in an inventory system when it takes time to collect, process, validate, and transmit inventory/demand data. A general framework is developed in this paper to describe information flows in an inventory system with information delays. We characterize the sufficient statistics for making optimal decisions. When the ordering cost is linear, the optimality of a state‐dependent base‐stock policy is established even when information flows are allowed to cross over time. Additional insights into the problem are obtained via a comparison between our models and the models with stochastic order lead times. We also show that inventory can substitute for information and vice versa.     

The Effects of Sharing Upstream Information on Product Rollover

The process of introducing new and phasing out old products is called product rollover. This paper considers a periodic‐review inventory system consisting of a manufacturer and a retailer, where the manufacturer introduces new and improved products over an infinite planning horizon using the solo‐roll strategy. We consider two scenarios: (1) the manufacturer does not share the upstream information about new‐product introduction with the retailer and (2) the manufacturer shares the information. For each scenario, we first derive the decentralized ordering policy and the system‐optimal ordering policy with given cost parameters. We then devise an optimal supply chain contract that coordinates the inventory system. We demonstrate that when the inventory system is coordinated, information sharing improves the performance of both supply chain entities. However, this may not be true if the inventory system is not coordinated. We also show that under the optimal contract, the manufacturer has no incentive to mislead the retailer about new‐product information in the information‐sharing model. When demand variability increases, information sharing adds more benefits to the coordinated supply chain. Our research provides insights about coordinating product, financial, and information flows in supply chains with product rollover.     

Production planning problem in manufacturing systems with general failure and repair time distributions

We study the optimal flow control for a manufacturing system subject to random failures and repairs. In most previous work, it has been proved that, for constant demand rates and exponential failure and repair times distributions of machines, the hedging point policy is optimal. The aim of this study is to extend the hedging point policy to non-exponential failure and repair times distributions and random demand rates models. The performance measure is the cost related to the inventory and back order penalties. We find that the structure of the hedging point policy can be parametrized by a single factor representing the critical stock level or threshold. With the corresponding hedging point policy, simulation experiments are used to construct input-output data from which an estimation of the incurred cost function is obtained through a regression analysis. The best parameter value of the related hedging point policy is derived from a minimum search of the obtained cost function. The extended hedging point policy is validated and shown to be quite effective. We find that the hedging point policy is also applicable to a wide variety of complex problems (i.e. non-exponential failure and repair times distributions and random demand rates), where analytical solutions may not be easily obtained.     

Static-dynamic uncertainty strategy for a single-item stochastic inventory control problem

We consider a single-stage inventory system facing non-stationary stochastic demand of the customers in a finite planning horizon. Motivated by the practice, the replenishment times need to be determined and frozen once and for all at the beginning of the horizon while decisions on the exact replenishment quantities can be deferred until the replenishment time. This operating scheme is refereed to as a “static-dynamic uncertainty” strategy in the literature [3]. We consider dynamic fixed-ordering and linear end-of-period holding costs, as well as dynamic penalty costs, or service levels. We prove that the optimal ordering policy is a base stock policy for both penalty cost and service level constrained models. Since an exponential exhaustive search based on dynamic programming yields the optimal ordering periods and the associated base stock levels, it is not possible to compute the optimal policy parameters for longer planning horizons. Thus, we develop two heuristics. Numerical experiments show that both heuristics perform well in terms of solution quality and scale-up efficiently; hence, any practically relevant large instance can be solved in reasonable time. Finally, we discuss how our results and heuristics can be extended to handle capacity limitations and minimum order quantity considerations.     

Unsold Versus Unbought Commitment: Minimum Total Commitment Contracts with Nonzero Setup Costs

We study a minimum total commitment (MTC) contract embedded in a finite‐horizon periodic‐review inventory system. Under this contract, the buyer commits to purchase a minimum quantity of a single product from the supplier over the entire planning horizon. We consider nonstationary demand and per‐unit cost, discount factor, and nonzero setup cost. Because the formulations used in existing literature are unable to handle our setting, we develop a new formulation based on a state transformation technique using unsold commitment instead of unbought commitment as state variable. We first revisit the zero setup cost case and show that the optimal ordering policy is an unsold‐commitment‐dependent base‐stock policy. We also provide a simpler proof of the optimality of the dual base‐stock policy. We then study the nonzero setup cost case and prove a new result, that the optimal solution is an unsold‐commitment‐dependent (s, S) policy. We further propose two heuristic policies, which numerical tests show to perform very well. We also discuss two extensions to show the generality of our method's effectiveness. Finally, we use our results to examine the effect of different contract terms such as duration, lead time, and commitment on buyer's cost. We also compare total supply chain profits under periodic commitment, MTC, and no commitment.     

Sourcing from Multiple Suppliers for Price‐Dependent Demands

We analyze a model that integrates demand shaping via dynamic pricing and risk mitigation via supply diversification. The firm under consideration replenishes a certain product from a set of capacitated suppliers for a price‐dependent demand in each period. Under deterministic capacities, we derive a multilevel base stock list price policy and establish the optimality of cost‐based supplier selection, that is, ordering from a cheaper source before more expensive ones. With general random capacities, however, neither result holds. While it is optimal to price low for a high inventory level, the optimal order quantities are not monotone with respect to the inventory level. In general, a near reorder‐point policy should be followed. Specifically, there is a reorder point for each supplier such that no order is issued to him when the inventory level is above this point and a positive order is placed almost everywhere when the inventory level is below this point. Under this policy, it may be profitable to order exclusively from the most expensive source. We characterize conditions under which a strict reorder‐point policy and a cost‐based supplier‐selection criterion become optimal. Moreover, we quantify the benefit from dynamic pricing, as opposed to static pricing, and the benefit from multiple sourcing, as opposed to single sourcing. We show that these two strategies exhibit a substitutable relationship. Dynamic pricing is less effective under multiple sourcing than under single sourcing, and supplier diversification is less valuable with price adjustments than without. Under limited supply, dynamic pricing yields a robust, long‐term profit improvement. The value of supply diversification, in contrast, mainly comes from added capacities and is most significant in the short run.     

The effects of component commonality in an infinite horizon inventory model

This paper studies the effects of component commonality in the context of an infinite horizon inventory model. Three models are proposed that are characterized by different degrees of component commonality. Assuming the three models all follow the same inventory policy, exact service level measures are derived and incorporated into cost optimization problems. With the infinite horizon assumption, potential setup cost reductions can be evaluated due to the inclusion of common components. The results indicate that, as expected, commonality incurs significant cost savings; what is new and unique is that setup cost may increase or decrease when commonality is present. In addition, when the behaviour of the optimal solutions is examined, it is found that some of the well-known properties suggested by the existing one-period models do not hold for this infinite horizon model.     

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.