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This paper treats a two-echelon inventory system. The higher echelon is a single location reffered to as the depot, which places orders for supply of a single com­ modity. The lower echelon consists of several points, called the retailers, which are supplied by shipments from the depot, and at which random demands for the item occur. Stocks are reviewed and decisions are made periodically. Orders and/or shipments may each require a fixed lead time before reaching their respective desti­ nations. Section II gives a short literature review of distribution research. Section III introduces the multi-echelon distribution system together with the underlying as­ sumptions and gives a description of how this problem can be viewed as a Markovian Decision Process. Section IV discusses the concept of cost modifications in a distribution context. Section V presents the test-examples together with their optimal solutions and also gives the characteristic properties of these optimal solutions. These properties then will be used in section VI to give adapted ver­ sions of various heuristics which were used in assembly experiments previously and which will be tested against the test-examples.



Some Modelling Theoretic Remarks on Multi-Stage Production Planning

Some Modelling Theoretic Remarks on Multi-Stage Production Planning

The papers of these proceedings are grouped according to the areas “Inventory and Distribution Problems”, “Multi-Stage Lot-Sizing”, and “Practical Applications and Hierarchical Integration Problems”. This grouping represents only one possible way of showing the relationships between the different contributions. In fact, there are many other alternative aspects which could lead to different groupings. Therefore, the following remarks may serve to exhibit some further relations. Of course, an exhaustive discussion and appreciation of all the different aspects the papers deal with is not intended.
Ch. Schneeweiss

Inventory — Production and Distribution Systems

Two-Stage Production Planning in a Dynamic Environment

In this document we report on research to develop and study mathematical models for production smoothing in a dynamic production environment. This effort was part of a larger study whose goal was to investigate the production planning practices for an electronic equipment manufacturing firm, and in particular to explore possible mechanisms for improvement. To motivate the presentation of our research, we indicate the nature of this production environment.
Stephen C. Graves, Harlan C. Meal, Sriram Dasu, Yuping Qui

Overview of a Stock Allocation Model for a Two-Echelon Push System Having Identical Units at the Lower Echelon

A two-echelon inventory system with one central warehouse and n identical regional warehouses is considered. Customer demand occurs only at the regional warehouses. A PUSH control system is used, implying that the allocation of stock is coordinated centrally. Given an initial system stock, a fixed planning horizon, and two shipping possibilities from the central warehouse until the next system replenishment (at the end of the planning horizon), the problem of deciding how much to ship initially to each warehouse is addressed. The stock retained at the central warehouse will be allocated to the branches at the second shipping opportunity so as to, if possible, balance the inventory levels, thus maximizing the customer service until the time of the next replenishment.
Given a desired customer service level, the aropriate initial system stock and the associated allocation are derived. The performance is canpared with a simple ship-all policy and with an extreme policy alla4ng a canplete redistribution of the inventory arorxg the regional warehouses at the second shipping opportunity. The results show that significant benefits can be achieved by the retention of a portion of the system stock at the central warehouse.
Henrik Jönsson, Edward A. Silver

System — Based Heuristics for Multi-Echelon Distribution Systems

A two-echelon distribution system is investigated. We determine the form and nature of the optimal ordering policy which is determined by means of Markovian Decision Process — analysis. Several heuristic procedures, all based on the echelon stock concept, are proposed and tested against the optimal solution in a simulation experiment.
R. Luyten

A Branch and Bound Algorithm for the Multi Item Single Level Capacitated Dynamic Lotsizing Problem

In this paper we present a branch and bound algorithm for the multi item capacitated lotsizing problem. The bounding procedure is based on a Lagrangean relaxation of the problem. The multipliers are updated using subgradient optimization.
Although this algorithm can solve the problem to optimality, it is mainly used as a heuristic. Extensive computational results are reported for a large number of problems.
Ludo F. Gelders, Johan Maes, Luk N. van Wassenhove

Multi-Stage Lot-Sizing

Aggregating Items in Multi-Level Lot Sizing

One way of reducing a multi-level lot sizing problem is to identify conditions on the cost parameters that will guarantee that a group of items can under all circumstances be produced at the same time. This means that this group of items can be treated as a single item. In this paper we present such conditions and discuss their applicability.
Sven Axsäter, Henry L. W. Nuttle

Optimal Lot-Sizing for Dynamic Assembly Systems

The simple case of a dynamic assembly system with no initial stocks is investigated as a test case for some new ideas on lot-sizing for Materials Requirements Planning. The model relies on the facility location formulation of Wagner’s and Whitin’s inventory problem. An iterative procedure is proposed that derives close upper and lower bounds through cost modifications based on Benders Decomposition and level-by-level optimization. The bounding procedure is integrated into a branch and bound scheme that outperforms by at least an order of magnitude its probably best competitor, the algorithm due to Afentakis et al (1984). The bounding procedure behaves excellently well as a heuristic and relates interestingly to the well-known lot-sizing methods of Graves (1981) and Blackburn and Milien (1982).
Kaj Rosling

Planning Component Delivery Intervals in Constrained Assembly Systems

Lot sizes for externally purchased components used in assemblies are often determined so that total inventory carrying costs and fixed ordering costs are approximately minimized. By focusing primarily on this criterion, a number of constraints relating to receiving dock capacity and storage area capacities are usually ignored. The goals of this paper are to develop a model and an algorithm that can be used to establish the time between external procurements and between movement of stock from stage to stage for each component in a multi-stage assembly system recognizing the presence of these capacity constraints.
John A. Muckstadt

Multi-Stage Lot-Sizing for General Production Systems

In this paper the multi-stage lot-sizing problem for general production structures is considered. General production structures are characterized by the fact that each stage may have several predecessor or successor stages. The objective is to minimize total costs which consist of a fixed charge per lot at each stage and linear holding costs. Time varying demand for final products is assumed to be known and has to be satisfied. A simple heuristic procedure is presented consisting of two phases. In the first phase a “basic policy” is determined which derives reorder times under the assumption of demand being constant. These reorder intervals are then in a second phase used to solve the time varying problem. In doing so a first possibility is to realize a cyclic policy simply according to the “basic policy”. A second possibility of taking into account non-stationarity is to take adjusted cost parameters and apply single-stage inventory models. A simulation study shows that considerable cost improvements can be realized using this heuristic.
Claus E. Heinrich, Christoph Schneeweiss

Practical Applications and Hierarchical Integration Problems

Practical Application of the Echelon Approach in a System with Divergent Product Structures

This paper deals with problems met in modelling a particular multi-echelon production and inventory situation in practice. These problems include process uncertainties, the influence of scheduling, a capacity-level decision and divergence in the last part of the product structure. First, echelon stocknorms are computed for the linear part of the product structure. These stocknorms are compared with the currently used stocknorms. To solve the last problem, the divergence, a simple heuristic is proposed.
K. van Donselaar, J. Wijngaard

Hierarchical Production Planning: Tuning Aggregate Planning with Sequencing and Scheduling

The first part of the paper describes the production planning problem of a major German food manufacturer and the hierarchical production planning system designed for solving it.
In the second part a major deficiency of Linear Programming models for aggregate planning of batch production is worked out. To remedy the defect the inclusion of ‘effective lot size demand’ within the Linear Programming model is proposed. Numerical results conclude the paper.
Hartmut Stadtler

The Design of an Hierarchical Model for Production Planning and Scheduling

In this paper, the design of an hierarchical model for production planning and scheduling is discussed in a real-life case study. The overall decision problem is partitioned into four levels: (1) aggregate production planning for product families, (2) detailed scheduling and sequencing, (3) determination of production orders for items, and (4) distribution and dispatching. Heuristic solution procedures are developed for all sub-problems. To respond to changes in input data, a rolling horizon procedure at the aggregate level and interactive replanning at all lower levels are suggested.
H. O. Günther


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