de Ag Ton Kok

Multi-echelon systems : a service measure perspective

This paper reviews the most important results on divergent multi-echelon systems. In particular, we concentrate on the interactions between the elements that constitute such a multi-echelon system, in order to determine several service measures (e.g. external customer service level and inventory holding costs). We distinguish between two types of policies: installation stock and echelon stock policies. A comparison between these two types of policies revealed that the complexity of the analysis is concentrated at different aspects, which are discussed by reveiwing the most important papers on both types. Special attention is given to the applicability of the models. Extensions to divergent multi-echelon systems with more than two stages are also treated.

Product recovery in stochastic remanufacturing systems with multiple reuse options

In many product recovery situations, returned products can be reused in multiple ways. Under these circumstances, the problem arises in which quantities reusable items should be allocated to the different remanufacturing options, especially in case of insufficient stock of returns. For this problem a periodic review model is formulated which also includes a disposal option and incorporates uncertainties in returns and demands for the different serviceable options. The structure of the optimal policy is analyzed, and it is shown that under specific allocation rules a near-optimal policy with a simple structure exists. An efficient computational procedure for determination of the optimal policy parameters is presented. This procedure is applied in a numerical investigation that gives interesting managerial insights into important product recovery issues.

Optimal control of a divergent multi-echelon inventory system

Consider a divergent multi-echelon inventory system, such as a distribution system or a production system. At every facility in the system orders are placed (or production is initiated) periodically. The order arrives after a fixed lead time. At the end of each period linear costs are incurred penalty costs are incurred at the most downstream facilities for back-orders. The objective is to minimize the expected holding and penalty costs per period. We prove that under the balance assumption it is cost optimal to control every facility by an order-up-to-policy. The optimal replenishment policy, i.e. the order-up-to-level and the allocation functions at each facility, can be determined by system decomposition. This decomposition reduces complex multi-dimensional control problems to simple one-dimensional problems.

Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach

We study a vehicle routing problem with soft time windows and stochastic travel times. In this problem, we consider stochastic travel times to obtain routes which are both efficient and reliable. In our problem setting, soft time windows allow early and late servicing at customers by incurring some penalty costs. The objective is to minimize the sum of transportation costs and service costs. Transportation costs result from three elements which are the total distance traveled, the number of vehicles used and the total expected overtime of the drivers. Service costs are incurred for early and late arrivals; these correspond to time-window violations at the customers. We apply a column generation procedure to solve this problem. The master problem can be modeled as a classical set partitioning problem. The pricing subproblem, for each vehicle, corresponds to an elementary shortest path problem with resource constraints. To generate an integer solution, we embed our column generation procedure within a branch-and-price method. Computational results obtained by experimenting with well-known problem instances are reported.

Linear programming models with planned lead times for supply chain operations planning

This paper contributes to the development of models for capacity constrained Supply Chain Operations Planning (SCOP). We focus on production environments with arbitrary supply chain structures. The demand for the end items is assumed to be exogenously determined. We solve the SCOP problem with Linear Programming models using planned lead times with multi-period capacity consumption. Using planned lead times increases the reliability of the communication between SCOP and Scheduling with regard to the feasibility of the planning. Planned lead times also reduce the nervousness in the system. We model capacity constraints on the quantity of items that can be assembled within a time interval. In particular, items can be assigned to multiple resources. We discuss two LP approaches which plan the production of items so that a sum of inventory costs and costs due to backordering is minimized.

Stock allocation in general multi-echelon distribution systems with (R,S) order-up-to-policies

In this paper we analyze stock allocation policies in general N-echelon distribution systems, where it is allowed to hold stock at all levels in the network. The goal is to achieve differentiated target customer service levels (fill rates). Various allocation rules and accompanying numerical methods that have already been developed for smaller networks are extended and compared in an extensive numerical experiment. We conclude that the extension of Balanced Stock rationing (see Van der Heijden (1996)) is the most accurate method, in particular in cases of relatively high imbalance. If the imbalance is not too high, the extension of Consistent Appropriate Share rationing (see De Kok et al., 1994; Verrijdt and De Kok, 1996) performs good as well.

Controlling a divergent 2-echelon network with transshipments using the consistent appropriate share rationing policy

Consider a two-echelon inventory system consisting of a central depot (CD) and a number of retailers. Only the retailers face customer demand. The CD is allowed to hold stock. In all stockpoints, the echelon inventory position is periodically raised to certain order-up-to-levels. At the central depot, incoming stock is allocated by using the consistent appropriate share rationing (CAS) policy. This means that this policy attempts to keep the ratio of the projected net inventory at any retailer over the system projected net inventory constant at any time. The size of this ratio depends on the customer service level every retailer requires, and the behaviour of the demand process. When the orders arrive at the retailers, an instantaneous rebalancing of the total net stock of the retailers takes place, so as to maintain all end stockpoint inventory at a balanced position. This rebalancing is realized by the transshipment of stock, assuming that the time to transship stock from one retailer to another is negligible compared to the replenishment lead time (lead time between CD and a retailer). Object of this analysis is the determination of all the control parameters (integral order-up-to-level, parameters of allocation policy at the CD and of the rebalancing policy at the retailer), so that the desired (different) service levels are attained at the retailers at minimal expected total costs. Exact expressions are developed to determine these parameters. However we will use some heuristics to actually compute these parameters, because of the intractability of the exact expressions. All analytical results are validated by Monte-Carlo simulation. The model developed will be compared with the same model without periodic, instantaneous rebalancing at the retailer. This yields insight into the conditions under which transshipment could be useful.

Distribution planning for a divergent n-echelon network without intermediate stocks under service restrictions

In this paper we discuss a periodic review control policy for general N-echelon distribution networks without batch size or capacity constraints. Only stockpoints at the end of the network are allowed to hold stock, whereas the intermediate stockpoints act as pure distribution centers that allocate incoming goods immediately to downstream stockpoints. Larger distribution networks (N = 3,4, 5) are often encountered in practice and therefore suitable inventory management policies are needed. Instead of defining a cost structure, we apply a service level approach where the main goal is to realize predetermined target service levels in the final stockpoints. A fast and accurate approximation method is presented which enables us to compute the parameters of the control policy; i.e. the system order-up-to-level and the allocation fractions for the allocation policy at the intermediate stockpoints. Finally, some attention is given to the important phenomenon of imbalance, which is caused by highly fluctuating demand processes at the final stockpoints. This phenomenon can affect the service performance of the developed control policy significantly.

Computational results for the control of a divergent N-echelon inventory system

Consider a divergent multi-echelon inventory system, e.g., a distribution system or a production system. At every facility in the system orders are placed (or production is initiated) periodically. The order arrives after a fixed lead time. At the end of each period linear costs are incurred at each facility for holding inventory. Also, linear penalty costs are incurred at the most downstream facilities for backorders. The objective is to minimize the expected holding and penalty costs per period. Within a class of practically useful policies the decomposition result is used to develop an algorithm which determines the control parameters of a near cost-optimal replenishment policy. A simulation study of a divergent three-echelon system reveals that this algorithm performs well.

Distribution planning for a divergent depotless two-echelon network under service constraints

In this paper we discuss a distribution planning procedure for a system consisting of one central depot supplying a number of end stockpoints. The central depot is not allowed to hold stock and allocates all incoming goods immediately to these end stockpoints. An ordering and allocation policy is presented which is based on a decomposition method. The emphasis lies on the realization of pre-determined target service levels at the end stockpoints. In this paper we present two adjustment methods which improve the service performance considerably in certain cases. Another important contribution of this paper is the generalization of the concept of imbalance. An analytical approximation of the probability of imbalance is presented. An extensive simulation study validates the analytical results.