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Quantitative models for inventory and production planning in closed-loop supply chains

The increased popularity of remanufacturing requires firms to rethink their supply chain systems and explore new avenues for the coordination of forward and reverse supply chains to transform existing supply chain systems into closed-loop supply chain (CLSC) systems. Several issues encountered in remanufacturing create unique operational challenges that must be taken into account in specifying guidelines for inventory and production planning (I&PP) for CLSC systems. These challenges provide a rich context for novel optimisation problems. Consequently, the literature on I&PP for CLSC systems has been growing continually. In this paper, we review the existing quantitative literature on I&PP for CLSC systems. We broadly categorise the present work into deterministic and stochastic problems according to the modelling of demand and return processes. Upon identifying the important relevant modelling parameters that influence the complexity of the underlying models, we provide a comprehensive classification of the current work according to these parameters. In addition, we highlight the solution methodologies used in the literature. Our goals are twofold: to provide a comprehensive exposition of the state-of-art in quantitative models for I&PP for CLSC systems, and to identify new avenues of research in I&PP that will aid in the practical implementation of remanufacturing.

A Dynamic Model for Inventory Lot Sizing and Outbound Shipment Scheduling at a Third-Party Warehouse

This paper presents a model for computing the parameters of an integrated inventory replenishment and outbound dispatch scheduling policy under dynamic demand considerations. The optimal policy parameters specify (i) how often and in what quantities to replenish the stock at an upstream supply chain member (e.g., a warehouse), and (ii) how often to release an outbound shipment to a downstream supply-chain member (e.g., a distribution center). The problem is represented using a two-echelon dynamic lot-sizing model with pre-shipping and late-shipping considerations, where outbound cargo capacity constraints are considered via a stepwise cargo cost function. Although the paper is motivated by a third-party warehousing application, the underlying model is applicable in the general context of coordinating inventory and outbound transportation decisions. The problem is challenging due to the stepwise cargo cost structure modeled. The paper presents several structural properties of the problem and develops a polynomial time algorithm for computing the optimal solution.

A comparison of outbound dispatch policies for integrated inventory and transportation decisions

This paper investigates the impact of alternative outbound dispatch policies on integrated stock replenishment and transportation decisions. The logistics literature reports that two different types of such policies are popular in current practice. These are time-based and quantity-based dispatch policies. Considering the case of stochastic demand, the paper presents analytical and numerical results showing that the cost savings obtained through quantity-based policies can be substantial. However, under a quantity-based policy, a specific delivery time cannot be quoted when the customer places an order. Hence, the paper also investigates the cost and customer waiting time implications of hybrid policies and demonstrates that hybrid policies are superior to time-based policies in terms of the resulting costs. Furthermore, although hybrid policies are not superior to quantity-based policies in terms of the resulting costs, they are superior in terms of a service measure which is quantified by the long-run average cumulative waiting time.

The Buyer-Vendor Coordination Problem: Modeling Inbound and Outbound Cargo Capacity and Costs

In this paper, we generalize the deterministic demand buyer-vendor coordination problem to simultaneously consider cargo capacity constraints and general inbound/outbound transportation costs. To this end, we first consider a replenishment cost structure that includes a fixed cost as well as a stepwise inbound freight cost for the vendor. We then extend our results to consider the case where both the vendor and the buyer are subject to this freight cost structure. Hence, in the second model, both the inbound and the outbound transportation costs/constraints of the buyer-vendor pair are modeled explicitly. For each case, we provide heuristic algorithms with promising error bounds. The error bounds for these heuristic methods are 6 and 25%, respectively. Using the costs of these heuristics as upper bounds, we also develop finite time exact solution procedures.

A Stochastic Model for Joint Inventory and Outbound Shipment Decisions

In this paper, we consider a vendor realizing a sequence of random order arrivals in random sizes. The vendor has the autonomy to hold/consolidate small orders until an economical dispatch quantity accumulates. Consequently, the actual inventory requirements at the vendor are in part determined by the parameters of the shipment release policy in use. In this context, we investigate the impact of shipment consolidation on the expected long-run average cost by simultaneously computing the optimal order quantity for inventory replenishment at the vendor and the optimal dispatch quantity for outbound shipments. Since we consider the case where demand follows a general stochastic bulk arrival process, obtaining exact analytical expressions for some key operating characteristics of the cost function is intractable. Hence, we provide easy-to-compute approximations which enable efficient numerical solutions for the problem. We also investigate: (i) the cases where consolidated shipments are preferred over immediate deliveries; (ii) the sensitivity of optimal integrated policy variables to demand/cost parameters; (iii) the potential savings that can be obtained by shipment consolidation; and (iv) the tradeoffs between the waiting time induced by shipment consolidation and costs saved. Our results provide insights into the impact of outbound transportation operations on inventory replenishment decisions and outbound distribution system design. Moreover, numerical testing suggests that significant cost savings (up to 57%) are possible with shipment consolidation.

Coordination of Inventory and Shipment Consolidation Decisions: A Review of Premises, Models, and Justification

This chapter takes into account the latest industrial trends in integrated logistical management and focuses on recent supply-chain initiatives that enable the integration of inventory and transportation decisions. The specific initiatives of interest include Vendor Managed Inventory (VMI), Third Party Warehousing/Distribution (3PW/D), and Time Definite Delivery (TDD) applications. Under these initiatives, substantial savings can be realized by carefully incorporating an out-bound shipment strategy with inventory replenishment decisions. The impact is particularly tangible when the shipment strategy calls for a consolidation program where several smaller deliveries are dispatched as a single combined load, thereby realizing the scale economies inherent in transportation. Recognizing a need for analytical research in the field, this chapter concentrates on two central areas in shipment consolidation: i) analysis of pure consolidation policies where a shipment consolidation program is implemented on its own without coordination, and ii) analysis of integrated policies where outbound consolidation and inventory control decisions are coordinated under recent supply-chain initiatives. The chapter presents a research agenda, as well as a review of the related literature, in these two areas. Some of the recent findings of the methodological research are summarized, and current and future research endeavors are discussed. By offering a theoretical framework for modeling recent supply-chain initiatives, the chapter highlights some of the many challenging practical problems in this emerging field.

A dynamic lot-sizing model with multi-mode replenishments: polynomial algorithms for special cases with dual and multiple modes

Abstract This paper generalizes the classical dynamic lot-sizing model to consider the case where replenishment orders may be delivered by multiple shipment modes. Each mode may have a different lead time and is characterized by a different cost function. The model represents those applications in which products can be purchased through various suppliers or delivered from a single source using various transportation modes with different lead times and costs. The problem is challenging due to the consideration of cargo capacity constraints, i.e., the multiple set-ups cost structure, associated with a replenishment mode. The paper presents several structural optimality properties of the problem and develops efficient algorithms, based on the dynamic programming approach, to find the optimal solution. The special, yet practical, cases of the two-mode replenishment problem analyzed in this paper are analytically tractable, and hence, the respective problems can be solved in polynomial time.

Integrated warehouse location and inventory decisions in a three-tier distribution system

A three-tier distribution network which consists of a single supplier at a given location, a single intermediate warehouse whose location is to be determined, and multiple retailers at given locations is examined. The problem is the integration of warehouse location and inventory replenishment decisions with the objective of minimizing the system-wide transportation and inventory-related costs. The case where the inventory replenishment decisions are coordinated using a power-of-two policy is considered and a mathematical model for the simultaneous computation of the warehouse coordinates and coordinated replenishment policy parameters is developed. The analytical properties of the integrated location-inventory model are characterized and efficient solution methods that rely on these properties are developed. Computational results demonstrating the performance of the proposed heuristic methods and the potential practical impact of integrated decision making for location and inventory decisions are reported. These results indicate that the proposed methods are capable of effectively producing high-quality power-of-two solutions within 6% of the lower bound for the instances tested. The presented analysis is also useful for identifying: (i) the level of interaction between these two types of decisions that are treated separately in the traditional literature; and (ii) problem settings where the integrated location–inventory model offers significant cost savings.

Integration of strategic and tactical decisions for vendor selection under capacity constraints

We examine a generalized vendor selection problem of a multi-store firm where the goal is the simultaneous determination of (i) the set of vendors the firm should work with and (ii) how much each store should order from the selected vendors. In addition to the typical costs associated with vendor selection and delivery between the vendors and their assigned stores, we explicitly consider the inventory-related costs and decisions of the stores. We emphasize the relationship between facility location applications and the problem at hand, and we propose an integrated vendor selection and inventory optimization model. Also, arguing that our model creates a venue for precise incorporation of realistic capacity constraints, we model throughput and dispatch capacities, explicitly. The model is a challenging mixed integer nonlinear program for which we propose an efficient solution approach that relies on Generalized Benders Decomposition (GBD).

Optimal myopic policy for a stochastic inventory problem with fixed and proportional backorder costs

In this paper, we consider a single product, periodic review, stochastic demand inventory problem where backorders are allowed and penalized via fixed and proportional backorder costs simultaneously. Fixed backorder cost associates a one-shot penalty with stockout situations whereas proportional backorder cost corresponds to a penalty for each demanded but yet waiting to be satisfied item. We discuss the optimality of a myopic base-stock policy for the infinite horizon case. Critical number of the infinite horizon myopic policy, i.e., the base-stock level, is denoted by S. If the initial inventory is below S then the optimal policy is myopic in general, i.e., regardless of the values of model parameters and demand density. Otherwise, the sufficient condition for a myopic optimum requires some restrictions on demand density or parameter values. However, this sufficient condition is not very restrictive, in the sense that it holds immediately for Erlang demand density family. We also show that the value of S can be computed easily for the case of Erlang demand. This special case is important since most real-life demand densities with coefficient of variation not exceeding unity can well be represented by an Erlang density. Thus, the myopic policy may be considered as an approximate solution, if the exact policy is intractable. Finally, we comment on a generalization of this study for the case of phase-type demands, and identify some related research problems which utilize the results presented here.