PROFIL 23

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Thi Thu Thuy Huynh

CAPACITY CONSTRAINTS IN MULTI-STAGE

PRODUCTION-INVENTORY SYSTEMS

– APPLYING MATERIAL REQUIREMENTS

PLANNING THEORY

PROFIL 23

PRODUCTION-ECONOMIC RESEARCH IN LINKÖPING

LINKÖPING 2006

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Production-Economcic Research in Linköping (PROFIL) Box 10034 S-58010 Linköping, Sweden

affiliated to

The Department of Production Economics Linköping Institute of Technology

S-581 83 Linköping, Sweden Tel: 013-28 10 00 ISBN 91-974579-2-2

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PREFACE

The objective of the PROFIL series is to provide information on research and other developments within the field of Production Economics by associates of Linköping Institute of Technology. Production Economics is basically an engineering discipline. It focuses on topics treating the interface between engineering and management and is therefore interdisciplinary in nature. It attempts at a two-way scientific integration; on the one hand to apply economic knowledge and reasoning to engineering and manufacturing problems, on the other, to utilise the arsenal of existing mathematical and quantitative engineering methods for analysing economic and managerial issues. It focuses on various allocation problems, in particular on the optimal use of productive resources within manufacturing and other industries. It encompasses developments in theory and application, wherever engineering and technology meet the managerial and economic environment in which industry operates.

From the economic science perspective, the methodological approach is based on the theory of production, a branch of microeconomic theory, and on methods of production and operations management. During the last several decades this theory has been integrated with mathematical and statistical models and methods from operational research and management science. The quantitative economic approach has thereby in a natural manner obtained a firm footing within the field – a development which has been strongly supported and encouraged by many Swedish industrialists and others.

Among the multitude of problems covered by this subject area can be found the development of principles for production planning and control on different levels and with varying scope. Rules for scheduling and sequencing work operations, the development of economic criteria for maintenance routines, principles for quality and inventory control etc. are other items of major interests, as well as methods, principles and procedures for investment planning and financial decisions, and also general theoretical frameworks. In this fascinating field of research, a host of interesting and intricate problems are awaiting solution, and their

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practical implementation may often be just round the corner. The field is also experiencing considerable attention today – from the academic as well as the practical point of view. The competitive edge of many world class companies is nowadays gained through manufacturing excellence, for instance from lead time and inventory reductions. Education and research in these areas are therefore important and rewarding, perhaps more so now than ever before.

This twenty-third publication in the PROFIL series entitled Capacity Constraints in Multi-Stage Production-Inventory Systems - Applying Material Requirements Planning Theory has been written by Thi Thu Thuy Huynh, who is presenting it as her doctoral thesis in Production Economics at Linköping Institute of Technology.

Linköping in January 2006

Robert W. Grubbström FVR RI Professor

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FOREWORD

I started my doctoral studies in February 2002 in the Department of Production Economics at Linköping Institute of Technology. While being a doctoral student for four years, I owe a lot to my teachers, family and friends.

First of all, I would like to express my gratitude to my supervisor, Professor Robert W. Grubbström, who has supported me throughout the whole process. He has contributed to many of the basic ideas of this dissertation. Without his guidance, I never could have finished this. I am greatly indebted to Professor Marija Bogataj and Professor Ludvik Bogataj, who have spent a lot of time with me for discussing suitable models and approaches. My thanks to Professor Ludvik Bogataj are not only for his valuable comments on my work, but also for his willingness to act as Faculty Opponent for this thesis.

My appreciation is also expressed to Professor Christopher O'Brien, Professor Lars-Erik Andersson and Professor Anders Segerstedt for their kindness in taking on responsibility as Committee Members.

At Billerud Skärblacka AB, I have received enthusiastic help from Ivo Fronzaroli, Marie Ardell and Mats Andersson. I greatly appreciate their help and the time they generously spent with me.

I would like to show my deep thanks to Docent Ou Tang who has supported me with several valuable pieces of advice in my work as well as on my conference trips.

My thanks go to all my colleagues in Department of Production Economics for making the department a nice place to work in.

Living in a foreign country, which is so far away from my home country, is not easy. I would like to thank the family of Son, Tuyet and Viet Ha,

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the family of Bo, Gabriella and Filippa, and the family of Märtha. They have given me a lot of help making my life easier and more enjoyable. I would also like to thank Lan, my room mate. We have spent a pleasant time together, which will be remembered all my life. Many thanks go to Phuong, who has helped me in programming the examples of this thesis. I also wish to thank several friends, whom I cannot list here, for sharing with me an interesting time in Sweden.

Finally, no word can express my deep gratitude to my parents, my sisters and my brothers for their love and for their encouragement throughout my long period of study.

Linköping in January 2006 Huynh Thi Thu Thuy

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1 INTRODUCTION

In a company, the role of production economics is to maximise the company’s economic result with respect to existing constraints. Production-inventory control is one of the important techniques of production economics.

1.1 Background and Motives

The objective of production-inventory control was stated by George W. Plossl (1985) in the following way:

“Production and inventory control is concerned basically with providing the information needed for the day-to-day decisions required to reconcile these objectives (maximum customer service, minimum inventory investment, and low-cost plant operation) in plant operations. The fact that these objectives are in basic conflict was readily apparent to the manager who owned his own small manufacturing company. He had invested his money in the machines and equipment in the plant, controlled his own manufacturing schedules and was his own sales representative. When a customer demanded immediate delivery, his alternatives was clear – either spend money on breaking into machine setups and working overtime or let the customer wait. He also had the alternative of carrying inventory in the future – finished products. Part or raw materials – so that he would be able to give his customer better service. The basic conflicts among these objectives existed in this one manager company and they were not easy to resolve, but at least the manger could see the conflict and weigh the alternatives.”

There are two techniques that a manufacturing enterprise may employ for purposes of inventory management (Orlicky 1975). The first one is stock replenishment, popularly known as statistical inventory control or the order point system. Usually, it is applied for single item models. Under this approach, the depletion in the supply of each inventory item is monitored and a replenishment order is issued whenever the supply drops to a predetermined quantity.

The second technique is Material Requirements Planning (MRP), which is used for a multi-level, multi-period production planning and control

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system. Multi-level production and inventory control is the subject of this work. It is commonly much more difficult to analyse and to control than a single-stage method.

MRP was introduced in the 1970’s. Much of the credit goes to three individuals by the names of Joseph Orlicky, George Plossl, and Oliver Wight. The American Production and Inventory Control Society (APICS), which was endorsed by Orlicky, Plossl, and Wight, was also credited with the introduction of MRP.

MRP is a system that controls inventory levels, plans production, helps to supply management with important information, and supports the manufacturing control system with respect to the production of assembled parts.

Figure 1.1 indicates an overview of how the MRP system works. The Product Structure File, the Master Production Schedule (MPS), and the Inventory Master File are the inputs of the MRP system. The output of the MRP system provides which component items that have to be produced and when, and this output can therefore be converted into the capacities required to produce these items.

The MRP system does not, itself, plan capacity requirements, but it provides an input to a capacity requirements planning system without which the latter cannot possibly function effectively (Orlicky 1975). In MRP systems, the MPS represent a plan for the production of all items over a given planning horizon. It specifies how much of each end-item that will be produced in each planning period, so that future component production requirements and material purchases can be calculated using the MRP component-explosion logic. As such, the MPS has to be feasible so that components can be produced within the capacity available in each time period.

Hence the production-inventory system is governed by the capacity constraints. Capacity constraints are considered in inventory planning for determining optimal target inventory positions.

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Product Structure File Master Production Schedule Inventory Master File Material Requirements Planning Manufacturing Orders Purchase Orders Various Reports

Figure 1.1. An overview of how the MRP system works.

Adopted from Foster (2002).

Capacity decisions are related to the performance of the system, e.g. the production lead time (Billington et al. 1983). Lead times are the times required to produce or assemble the items.

The study of the MRP system has received a great deal of attention and extensions have been made to connect these studies with other theory. With the objective of obtaining optimal solutions, quantitative aspects of planning and-inventory control have resulted in several pieces of theoretical work on MRP and similar multi-level production-inventory systems. One breakthrough in this direction is the application of Laplace transform and Input-Output Analysis methodology.

Combining

Laplace transforms

and Input-Output Analysis has proved to be advantageous for dealing with complex production-inventory systems (Grubbström and Tang, 2000).

The useful relationship between the Laplace Transform and the Net Present Value (NPV) has been known for a considerable time

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(Grubbström, 1967) and the NPV principle, considered as a more correct approach than the average cost measure, is being applied to production-inventory theory in a rapidly increasing number of articles (Grubbström and Lundquist 1977, Grubbström 1980, Thorstenson 1988, Grubbström and Jiang 1990). Here, we continue with this combined methodology. The Input-Output methodology is used to generalise the product structure as well as the timing relationship between items. The fundamental equations of the production-inventory system are described by the combination of transforms and Input-Output Analysis (Grubbström and Ovrin 1992, Grubbström and Molinder 1994). These equations are balance equations in the frequency domain explaining the development of total inventory, available inventory, backlogs and allocations (items reserved for production). The combined methodology of Input-Output Analysis and Laplace (or z-) transforms applied to production-inventory systems, we name MRP theory. More recent outcomes are the dissertations by Molinder (1995), Tang (2000), and by Wang (2002). Different directions of research within MRP theory have been developed, such as by Horvat and Bogataj (1996), Martin (1997), Bogataj (1999), Bogataj and Bogataj (2004).

This work treats the capacitated multi-level, multi-stage production inventory problems within MRP theory. Input-Output Analysis and Laplace transforms or z-transforms are then employed as the essential tools.

1.2 Objectives

In this thesis, capacity-constrained aspects of multi-level, multi-stage production planning are investigated. The aim has been to extend Material Requirements Planning Theory (MRP Theory) to cover more general problems dealing with capacity constraints, in particular when non-zero lead times are present and the processes take place in continuous time.

The methodology used in this book is the quantitative method, which is based on the principles of Operational Research and Management

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Science. The models are built on mathematically oriented production planning and inventory theory.

In previous studies, the NPV objective function of multi-item production- inventory system has been developed. The optimisation conditions and capacity constraints have been generated. The problem is easy to formulate but difficult to solve for the optimal solution, especially when lead times are non-zero and external demand is considered in continuous time.

Our aim is to extend our study to situations explaining how production-inventory system should works under capacity constraints when lead times are non-zero. First we develop a heuristic approach to get an inaccurate, but simple, result. Then, we state optimisation approaches for the models in several steps.

This thesis also examines the fundamental equations of MRP Theory in order to analyse the possibility to obtain closed-form expressions for the time development of the system, when standard ordering policies of MRP are applied. Input-Output Analysis is used for capturing the Bill of Materials (the product structure) and Laplace transforms for describing the advanced timing requirements (lead times).

1.3 Limitations

Multi-level production and inventory control causes several problems and difficulties in practice. Hence we have simplified the models with some assumptions.

The product structures which are used in our work are assembly type (left part of Figure 1.2) only. Other types of production such as arborescent structures (right part of Figure 1.2) are not treated in this thesis. The assembly type of product structure is the prevailing type in manufacturing industries, whereas extraction industries, such as mining, often have product structures of the arborescent type. In various process industries there are often different mixtures of these types of structures. In a sequential structure, each item produced has only one predecessor

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and only one successor. A sequential structure is therefore both of an assembly and an arborescent type.

Assembly structure Arborescent structure Figure 1.2. The two basic types of product structures.

End products are represented by top-level boxes, and subassemblies or semi-processed products by lower-level boxes.

We focus on the multi-period production planning and control system applying MRP theory. Other alternative techniques such as inventory control and order point systems is not the subject of this work.

We used the NPV instead of total or average cost as our objective functions. NPV is useful when we apply the Laplace transform.

The only “uncertain” aspect considered here is stochastic demand. We mostly assume that lead times are given constants, even if lead times often can be stochastic in the real system. We also ignore other kinds of uncertainty such as machine reliability and production costs changing over time.

Other limitations will be explained in the chapters to follow. Often such limitations are consequences of assumptions set forth.

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1.4 Outline and Scientific Contribution

As mentioned above, the multi-stage, multi-level capacity-constrained production-inventory problems have been treated in some previous studies. In Table 1.1, we have described the contribution from previously completed studies and the current contributions from this dissertation. The systems considered are classified with respect to properties of demand, properties of the time scale and with respect to whether or not the lead times are zero (or unity) or not in discrete time, or zero or non-zero in continuous time. The papers relevant to this development are listed in Table 1.1.

Discrete time Continuous time

Deterministic demand Varying τ Grubbström- Huynh (2006) τ ≤ 1 Segerstedt (1996) Varying τ Grubbström- Huynh(2005a) τ = 0 Stochastic demand Varying τ τ ≤ 1 Grubbström-Wang (2003) Varying τ Huynh (2006) τ = 0

Table 1.1. Contributions to the development of multi-stage, multi-level capacity-constrained production-inventory theory with respect to

demand, time framework and lead times τ.

Segerstedt (1996) built a dynamic programming model to solve the capacitated multi-level, multi-stage production-inventory problem under the assumption that lead times either all are zero or all have a length of one period in discrete time. Later, Grubbström and Wang (2003) presented a model in discrete time when external demand is stochastic

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and production lead times are zero. From 2002 until now, we develop the other cases in turn.

This thesis consists of nine chapters. This first chapter is an introduction to the report. Here is presented the background and motives, the objectives, the limitations and the general contents of other chapters. Chapter 2 discusses the major research framework of this work. It also presents a literature review on the topics of multi-stage production inventory planning which focuses on capacity constraints. Our basic model for multi-level, multi-stage production inventory is also presented in this chapter for providing the general formulas of the model and the notation applied.

Chapter 3 examines the fundamental equations of MRP Theory. These equations are balance equations in the frequency domain explaining the development of total inventory, available inventory, backlogs and allocations (items reserved for production). The objective is to analyse the possibility to obtain closed-form expressions for the time development of the system, when standard ordering rules of MRP are applied. Standard ordering policies presented are Lot-For-Lot (L4L), Fixed Order Quantity (FOQ), Fixed Period Requirements (FPR) which are mostly used when demand is considered to be deterministic. However, this chapter also analyses aspects of the Economic Fixed Order Quantity policy and the Optimal Fixed Period Requirements when assuming demand to be stochastic. The approaches analyse the problems both in the time domain and in the frequency domain. This chapter is based on Grubbström and Huynh (2005b).

In Chapter 4 we present a heuristic methodology for solving a capacity constrained production-inventory model. This chapter extends the Grubbström and Wang (2003) model to the case when lead times are non-zero. It applies the heuristic method of França et al (1996) to solve lot-sizing in a multi-stage system. Firstly, the uncapacitated model is solved by dynamic programming to get an initial solution. If the initial solution is capacity infeasible, a smoothing procedure and an improvement procedure are applied to obtain feasible solution. This gives an inaccurate, but simple solution. The contents of this chapter are rewritten from Grubbström and Huynh (2002).

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From Chapter 5 to Chapter 7, we solve the capacitated level, multi-stage problems. The Laplace transform together with Input-Output Analysis are employed as tools for modelling the production-inventory system. This approach has been successfully used for formulating a theoretical base for Material Requirements Planning. The development follows the steps described below.

In Chapter 5, we assume that demand is deterministic. The model is considered in discrete time and production lead times are non-zero. A dynamic programming model is formulated for the purpose to solve the optimisation of our objective function. The objective function is the NPV in this and the two following chapters. Here is the first time we introduce an efficient state space for dynamic programming, in which historical values of cumulative production and cumulative demand are given state variables. This chapter is based on Grubbström and Huynh (2006). In Chapter 6, we turn to continuous time theory and develop the corresponding methodology for the case that lead times are non-zero, while keeping external demand deterministic. Our emphasis is on the design of the state space, the properties of which depend on the product structures (the input matrix), the distribution of lead times among the production processes (the lead time matrix), and on the historical sequences of production and demand vectors. Once an efficient state space is designed, dynamic programming once again is applied as the solution method. The Net Present Value Principle is applied. This chapter is based on Grubbström and Huynh (2005a).

In Chapter 7, we cover a model with stochastic demand in continuous time. The lead times are allowed to be non-zero. This is the most general system treated, and all other theory referred to, may be considered as special cases hereof. In Table 1.1, the thick-line triangle represents this theory. Appropriate states for applying dynamic programming are also designed. Once again, the Net Present Value is applied. The Laplace transform together with Input-Output Analysis have been employed as tools in constructing the model. The content of this chapter comes from Huynh (2006).

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In addition, in Chapter 8, the production planning procedures applied in a paper mill have been studied, and they are here presented in the form of a case study. We investigate the production planning and related capacity-constraints of this plant.

Chapter 9 summarises our work and states some of our conclusions. There we have also listed some further issues suggested for future research.

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2 LITERATURE REVIEW AND RESEARCH

FRAMEWORK

Production planning has a fundamental role in any manufacturing operation. The problem is to decide what type of product, and how much of each product that should be produced in future time periods. The decisions should be based on many factors, including period machine capacity, production lead times, holding costs, etc. Of primary importance is the estimate of demand for manufacturer's products in upcoming periods.

Multi-stage production planning and inventory control is one of the most challenging problem areas in the field of manufacturing management (Axsäter et. al., 1986). One of the problems concerned is the application of MRP Theory to multi-stage lot sizing. MRP should be ranked at the top for analysing multi-level production-inventory control (Axsäter and Rosling, 1994). The widespread and popular use of MRP systems in industry has also resulted in an increased interest in the topic of decision making in capacitated multi-stage production-inventory system.

This chapter provides a literature review of work treating capacitated multi-stage production-inventory systems and the application of MRP theory and Input-Output Analysis together with Laplace transforms to production-inventory systems.

2.1 Capacity-Constrained Multi-Stage Production

Planning and Inventory Control

We structure this section in the following areas:

- Deterministic demand, multi-stage production planning and inventory control,

- Stochastic demand, multi-stage production planning and inventory control.

- Multi-stage production planning and inventory models applied in industry.

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2.1.1 Deterministic Demand

Throughout this literature review of deterministic demand models, only discrete-time approaches have been found. Billington et al (1983) modelled the general MRP problem with capacity constraints. They presented an integer linear programming model of a general capacity-constrained multi-level MRP system. They limited themselves to very few constrained facilities (one or two), since they used a simple heuristic optimisation method. In 1994, also Tempelmeier and Helber used a heuristic for solving multi-item, multi-level lot-sizing problems with multiple capacity constraints.

França et al (1996), have used a heuristic method for solving lot-sizing in multi-stage systems. First, they used the Wagner-Whitin dynamic lotsizing algorithm for each item of the system. Subsequently, a smoothing procedure and an improvement procedure are introduced for finding feasible capacity solutions.

Segerstedt (1996) dealt with an optimisation approach for the multi-level, multi-stage capacity problem. Demand was assumed to be deterministic and the discrete production lead time either zero or unity. Dynamic programming was applied to find the optimum solution using the NPV as objective function.

Ip et al. (2000) solved the earliness and tardiness production scheduling and planning problems. They applied a genetic algorithm in order to obtain an optimal solution to the multi-product production environment. Their objective was to find an optimal lot-size production schedule within a production planning and scheduling horizon such that the total cost of earliness and tardiness penalties could be minimised while satisfying the manufacturing process capacity constraints. Both a genetic and a linear programming approach were used and compared.

A heuristic genetic algorithm for capacitated lot-sizing problems was developed by Xie and Dong (2002). With a zero lead time assumption, they built a model to minimise the total cost for a multi-stage production-inventory model. Their product structure could be a single level system, or a serial or an assembly system, and other variants. The capacity constraints could be of several types: uncapacitated, capacitated

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single resources and/or capacitated multiple resources. It should be noted that the cost of computations and computation time for this algorithm is significant, since the complexity of problem is very high.

Recently, Clark has built some models for capacity-constrained multi-level MRP systems with deterministic demand. In Clark (2003), three mixed integer programming (MIP) models and solution methods were developed to assists in identifying a capacity feasible MPS in MRP systems. An optimisation of capacity utilisation model was formulated and then solved using two approaches based on model approximation and sequential decomposition. The objective function applied was the traditional cost principle. Lead times were here assumed to be zero. In contrast, not avoiding lead times, Clark and Armentano in their report (1993) explicitly dealt with lead times in their multi-stage lot sizing problems treated.

The limited capacity manufacturing system also was treated by Bouhia and Abernathy (2003). In this paper, they defined a list of ordering and scheduling policies in a multiple product, multiple plant case to maximise profits subject to capacity limits of the system. Their solution method was based on simulation methodology.

Pongcharoen et al. (2002) introduced a genetic algorithm for the finite capacity scheduling of complex products with multiple levels in the product structure. Their algorithm including problem encoding, chromosome representation and initialisation, genetic operation, repair process, fitness measurement and chromosome selection is described in detail. A statistical analysis showed that the size of the problem has a major influence on the penalty costs concerned.

Jolayemi and Oloruniwo (2004) developed a deterministic model for planning production and transportation quantities in a multi-plant and multi-warehouse environment, where opportunities to increase capacity levels were available. A mixed integer linear programming model was applied to analyse this model.

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2.1.2 Stochastic Demand

In practice, almost every production-inventory is subject to influences with uncertainty. This may be external demand, the supply lead time or other operational parameters (Tang, 2000). Here, we will be dealing with the uncertainty of external demand only.

Albritton, M., et al (2000) considered a multi-product system in a single stage model, each resource having a finite capacity. The demand for each product was stochastic with log-normal distributions. When demand for a product exceeded the available inventory for that product, the manufacturer would lose sales. They solved this problem with two variants of Monte Carlo sampling-based optimisation techniques, to which they refer as simulation-based optimisation methods. However, they did not consider any multi-stage aspect.

Grubbström and Wang (2003) have developed a capacity-constrained multi-level multi-stage production-inventory system with stochastic demand and zero lead times. Their optimisation methodology was dynamic programming. In their results stated, a comparison was made between the capacitated and uncapacitated problems indicating the value of adding new capacity.

Brander et al (2005) have presented a simulation study under stationary stochastic demand. They used a heuristic approach to calculate the lot size for the economic lot scheduling problem. In this study, the actual demand had a uniform distribution. The purpose of this paper was to examine if a deterministic model could be used when demand has a stationary stochastic distribution. The authors concluded that the model used for determination of lot size was of lesser importance than the decision rule used for identifying the item to produce and, in particular, when to produce it.

2.1.3 Models Applied in Industry.

There are some papers dealing with multi-product multi-period problems in industrial systems. Yan (2000) presented a hierarchical stochastic production planning for flexible automation workshops. Their software

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package, named as stochastic interaction/ prediction algorithm, was developed as a result of this study.

A capacity planning system that considered both the capacity and the capability of equipment was developed for multi-product semiconductor fabs (Chen et al, 2005).

Recently, Mula et al (2006) presented a new linear programming model for medium term production planning in a capacity constrained MRP, multi-product, multi-level and multi-period manufacturing environment. This paper provided three fuzzy models with flexibility in the objective function. The paper demonstrated the usefulness and significance of MRP modelling with flexible constraints under uncertainty in demand. Their model was tested using real data from an automobile seat manufacturer.

2.2 The Application of MRP Theory Employing

Laplace Transform and Input-Output Analysis to

Multi-Level Multi-Stage Production and Inventory

Systems

The idea of applying Input-Output Analysis to MRP and multi-level production and inventory systems has been studied for some years in the Department of Production Economics at Linköping Institute of Technology.

In Grubbström and Lundquist (1989), the relationships between Input-Output Analysis, MRP and production function were discussed. They concluded that there was an obvious relationship between master production scheduling, MRP, the bill of material (BOM), manufacturing resource planning (MRP II), and a general linear production-economic system interpreted in terms of an input-output model.

The useful relationship between the Laplace Transform and the Net Present Value (NPV) was presented by Grubbström (1967). Moreover, the NPV principle, considered as a more correct approach than the average cost measure, has been applied to production-inventory theory in

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an number of articles (Grubbström and Lundquist 1977, Grubbström 1980, Thorstenson 1988, Grubbström and Jiang 1990), Van der Laan (2003), and Teunter (2003).

Applying the combination of transforms and Input-Output analysis to production-inventory was presented in Grubbström and Ovrin (1992) and Grubbström and Molinder (1994), and was further developed in Molinder (1995). In Grubbtröm and Ovrin (1992), the theory was originally introduced using the z-transform for application in discrete time. In Grubbström and Molinder (1994) and Molinder (1995), the theory was converted from discrete time to continuous time by introducing the Laplace transform. The advantages from combining the Laplace transform and Input-Output Analysis when dealing with complex production-inventory systems have been argued in Grubbström and Tang (2000).

Grubbström and Tang (1997) have reviewed some early writings related to the topics of MRP and Input-Output Analysis. They also presented simple examples to further explain the interrelationship between these two methodologies.

Recently, a few publications on capacitated multi-stage production-inventory applying MRP theory have appeared (Grubbström and Wang (1999), Grubbström and Wang (2000a), Grubbström and Wang (2000b), and Grubbström and Wang (2003)). There are also other recent outcomes such as the theses of Molinder (1995), Tang (2000), and of Wang (2002).

2.3 Capacity-Constrained Multi-Level Multi-Stage

Production Inventory Systems – Some Notation and

Explanations.

We now describe the capacity-constrained multi-level multi-stage production inventory systems. Our basic notation is introduced for the application in all chapters to follow.

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T time until given horizon.

N number of all items in the system. N’ number of end items

M number of resources constraining production. s complex Laplace frequency.

i

n number of set-ups for the production of batches of item i.

ij

t time at which batch no j of item i is completed (decision variable), i = 1, 2, ... , ni.

τ_i lead time for the production of item i.

τ( ) s (N×N)-dimensional lead time matrix with a lead time operator _esτi_{in its ith diagonal position and zeros}

elsewhere. ij

ν jth set-up of the production of item i referred to completion time tij. As a time function νij may be written as a sequence of Dirac impulses (δ t−t_ij), j = 1, 2, ... ni. ( )s

ν vector of all completion times in Laplace transform notation

i

K set-up cost for producing a batch of item i,.

ρ _{continuous interest rate per time unit.}

ij

h′ number of units of item i required to produce one unit of its immediate successor j. Collected into the input matrix

. ′

H kj

h′′ capacity of resource k needed to produce one unit of item j collected into the input matrix H . ′′ H and ′ H make up ′′

the joint input matrix H.

ri sales price per unit of end product i collected into the row

vector r.

ci echelon production cost per item i collected into the

column vector c.

wk variable cost for using resource k per time unit collected

into the column vector w.

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Di external demand rate of item i at time t.

( ) i

D t external cumulative demand per time unit of item i from

time 0 to t, 0 ( ) ( ) t i i D t =

∫

D α αd .

Pij number of units of item i (lotsize of batch no j) completed

at time t_ij. ( )

i

P t cumulative number of units of item i produced from time 0 to t.

Bi(t) level of stockouts of item i at time t.

Ri(t) Available inventory of item i at time t.

NPV Net Present Value, objective function. E (·) expectation operator.

( )

x t short-hand notation for a time derivative of x(t).

Tildes or the operator £

{ }

⋅ are used to denote Laplace transforms of the corresponding time functions, i. e. f s( )=£

{

f t( )

}

=

0 ( ) st f t e dt ∞ −

∫

=

0 n st n n f e ∞ − =

∑

for a continuous ( )f t or a discrete time function f , n respectively, and bars for cumulative functions, like

0 ( ) ( ) t f t =

∫

f α αd or 0 n n k k f f = =

∑

.

In Grubbtröm (1999), an NPV approach for the general model of multi-level multi-stage production-inventory problem was presented. The following expression is the NPV of the cash flow,

(

)

(

)

(

)

(

)

E NPV E ( ) E ( ) E ( ) T T e ρ ρ ρ ρ − =r D − B − B ( ) ( )ρ τ ρ ν ρ( ) ( ) − +c w P −K

, (2.1)

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where r and K are N-dimensional row vector containing the components rj (sales prices) and Kj (set-up costs), respectively, and (c +

w)

an

(N + M)-dimensional row vector with components ci + wi. The lead time

matrix τ

( )

ρ appearing in the last term accounts for allocating the set-up costs to the beginning of the lead time, since ( )ν ρ is defined from the completion dates. The production costs have been allocated to the completion dates by using the (N + M)-dimensional column production vector ( )P ρ .

In (2.1), E

(

D( )ρ

)

is an N-dimensional column vector of component presenting expected external demand discounted to time zero with no consideration to stockouts, ρE

(

B( )ρ

)

is a similar vector of discounted changes in backlogs, and E

(

B( )T

)

e−ρTcontains correction terms for the sake of backlogs not being satisfied at the horizon T, since the process then terminates and no more production is planned.

To get the optimal solution, we search for the maximal value of (2.1) subject to the following constraints.

The available inventory R( )s is cumulative production ( )P s less cumulative demand ( )D s and cumulative internal demand

( ) ( )s s

Hτ P and must be non-negative. (0) (0) ( ( )) ( ) ( ) ( )s s s s s + ⎡ − + − ′ − ⎤ = ⎢ ⎣ ⎦ R B I H τ P D R _{⎥ , (2.2)}

where

[ ]

· + is the maximum operator Max{0, }⋅ , is the total production of all items and

( )

s

P

( ) ( )s s

Hτ P is the internal demand. In (2.2), all vectors are N-dimensional

If external demand cannot be satisfied, we assume that this demand is backlogged and satisfied at the time available inventory starts to become positive one again.

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(0) (0) ( ( )) ( ) ( ) ( )s s s s s + ⎡ − − − ′ + ⎤ = ⎢ ⎣ ⎦ B R I H τ P D B _{⎥ (2.3)}

Regarding end items (the N’ first components), the non-negativity is taken care of by the backlogging opportunity.

For lower-level items, the backlogs are only allowed for external demand, but internal demand must always be met in order for the production plan to be feasible (the available inventory constraint):

{ }

1 1 (0) (0) ( ( )) ( ) £ ( )s £ s s s − ₌ − ⎧ − + − ′ ⎫_≥ ⎨ ⎩ ⎭ R B I H τ P R _⎬ 0, (2.4)

where _£−1

{ }

⋅ denote the inverse Laplace transform.

The second constraint is that of the capacity limitations,

1 ( ( ) ) ( ) £ ( )s s s s − ⎧ ₋ ′′ − ⎫ ⎨ ⎩ ⎭ H τ I P C _⎬≥0. (2.5)

In the next chapters, we will search for the optimal solution to (2.1) subject to constraints (2.4) and (2.5) following several steps of development.

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3 ANALYSIS OF STANDARD ORDERING

POLICIES WITHIN THE FRAMEWORK

OF MRP THEORY

In MRP theory the time development of the production-inventory system is determined by a set of fundamental equations on the one hand, and external demand and production decisions, on the other. The fundamental equations are balance equations applicable to available inventory, total inventory and backlogs, using Input-Output Analysis for capturing the Bill of Materials and Laplace transforms for describing the advanced timing requirements (lead times).

This chapter aims to examine the fundamental equations of MRP Theory into in order to obtain closed-form expressions for the time development of the system, when standard ordering policies of MRP are applied. The fundamental equations of MRP Theory have been developed in several earlier papers, beginning with Grubbström and Ovrin (1992) and in some earlier unpublished studies. These equations are balance equations in the frequency domain explaining the development of total inventory, available inventory, backlogs and allocations. Input-Output Analysis is used for capturing the Bill of Materials and Laplace transforms for describing the advanced timing requirements (lead times) In Grubbström and Tang (2000) an overview is presented. This chapter analyses the question of developing closed-form expressions for production when applying basic ordering rules. Order sizes are to be decided in time when inventories are zero or near to zero.

In Grubbström and Ovrin (1992) the problem treated here was touched upon. In that paper the processes took place in discrete time, and the z-transformation was applied. Grubbström and Molinder (1994) and Molinder (1995) also followed up some research into this issue in the continuous time case.

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In Orlicky (1975), some standard ordering policies such as Lot-For-Lot (L4L), Fixed Order Quantity (FOQ), Fixed Period Requirements (FPR) are defined.

The L4L rule, being the simplest (also called “as required”) involves an order to be placed exactly large enough to cover requirements. This means that available inventory is kept at a zero level (assuming no initial inventories). In L4L, both the size of the order and the interval between orders will vary over time in a general case. Production, by this rule, is therefore adapted directly to requirements.

With the FOQ policy, the order size is always the same. An order is placed as soon as there is not enough available inventory to cover requirement. If demand fluctuates, the interval between orders also must fluctuate. In general, available inventory will only reach a zero level on occasion.

Applying the FPR policy, the interval during which total demand is generated and which the order should cover is constant. An order is placed just large enough to cover the total requirements during each such interval. At the end of these intervals, available inventory will have dropped to zero.

We present a brief overview of the fundamental equations of MRP Theory in Section 3.1. This is followed by deriving the basic properties of production for the three ordering policies in relation to this theory in Section 3.2. Sections 3.3 and 3.4 are devoted to solving for the production development. In Section 3.5, the optimal FOQ and optimal FRP are presented when demands are stochastic.

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3.1 The

Fundamental

Equations of MRP Theory

3.1.1 Definitions

Figure 3.1. Flowchart of the fundamental equations of MRP Theory (Grubbström and Tang 2000).

The fundamental equations of MRP Theory are balance equations describing the time development of total inventory, available inventory, backlogs, and allocations (see Grubbström and Tang, 2000). With the policies we are studying, backlogs will not occur in deterministic demand cases. In such cases, the fundamental equations for total inventory and available inventory may be written:

(

)

(0) ( ) ( ) ( )s s s s + − − =S I H P D S _, _(3.1)

(

)

(0) ( ) ( ) ( ) ( )s s s s s + − − =R I Hτ P D R _{, (3.2)}

where is the column vector of items in inventory (including allocations), is the column vector of items in available inventory (total inventory less allocations), is the vector

S R

D of external demand, is the vector of items produced, H is the input matrix (the Bill of

Materials), and I is the identity matrix. The lead time matrix

P

( )s

τ is a matrix with lead time operators in its diagonal positions:

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1 2 0 0 0 ( ) 0 0 N s s s e e s e τ τ τ ⎡ ⎤ ⎢ ⎥ ⎢ =_⎢ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ τ " " # # % # " 0 ⎥ ⎥. (3.3)

Internal (dependent) demand is given by Hτ( ) ( )s P s , since the input matrix and lead time matrix together determine all advanced requirements of sub-components in amount and timing, given a production schedule ( )P s .

The main problem treated in this chapter, is how production P is

determined by each of the policies, when facing a given external (independent) demand .

D

In one-item systems, there is no internal (dependent) demand and, thus, independent and dependent demand, on the one hand, and available and total inventory, on the other, will coincide. The fundamental equations (1)-(2) then collapse into:

(0) ( ) ( ) ( ) R P s D s R s s + − = _. _(3.4)

3.1.2 Properties of Cumulative Requirements

The minimum necessary production to meet external (independent) requirements is always given by

(

)

1

_{( ) ( )}

2 3 ( )s ... ( )s − ⎛_⎜ ⎝ − = + + + + I Hτ D I Hτ Hτ Hτ ⎞⎟_⎠D , (3.5)

where the Neumann expansion has been used. This expansion is valid as long as the numerical values of all characteristic roots of H are less than

unity. For assembly systems, in which H is triangular with zeros along

its main diagonal and above, this is indeed so. The expansion will converge in at most N terms, where N is the dimension of H.

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Assuming only one end product at the top level, we have 1 0 ( ) 0 D s ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ D # , (3.6)

and minimum cumulative requirements become

(

)

1

( ) ( )

2 3 1 1 1 1 / ... / col col D s D − ⎡ ₋ ⎤ ₌_⎡ ₊ ₊ ₊ ₊ _⎤ ⎢ ⎥ ⎣ ⎦ ⎣ I Hτ ⎦ I Hτ Hτ Hτ s (3.7) The triangular nature of H for an assembly system creates the following

first elements

(

)

1 1 1 1 / / row D s D s − ⎡ ₋ ⎤ ₌ ⎢ ⎥ ⎣ I Hτ ⎦ (3.8)

(

)

1 1 1 21 2 / s / row 1 D s e H D sτ − ⎡ ₋ ⎤ ₌ ⎢ ⎥ ⎣ I Hτ ⎦ (3.9)

(

)

1 1

(

2

)

1 13 32 21 3 / s s / row 1 D s eτ H H H eτ D s − ⎡ ₋ ⎤ ₌ ₊ ⎢ ⎥ ⎣ I Hτ ⎦ (3.10)

(

)

1 1 4 / row D s − ⎡ ₋ ⎤ ⎢ ⎥ ⎣ I Hτ ⎦

(

3 2 3

)

1 2 ( ) 14 42 21 43 31 43 32 21 1/ s s s s eτ H H H eτ H H eτ H H H e τ τ+ D s = + + + _(3.11)

This reveals that requirements occur at times in advance of external demand in such a way that steps are generated sums of lead times ahead of top-level requirements. The further down in the product structure tree, the more opportunities for additional steps exist. For instance, on level 4, if all relevant

H

_ij are non-zero, then, generated by an external demand

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event at , items will be required at the points in time t t− , τ₁ t− − , τ τ1 2 1

t− − , and τ τ₃ t− − − , whereas on level 3, items may be required τ τ₁ ₂ τ₃ only at t− and τ₁ t− − . The total number of possible times τ τ₁ ₂ increases geometrically, but all need not exist. One may also note that all possible times at a higher level are repeated again at all lower levels. Therefore the set of possible times for the entire system can be found by studying the lowest level in the system. But zeros in the matrix below the diagonal will rule out some combinations.

A sequence of external demand events together with the set of lead times will thus generate a sequence of possible internal demand events. On levels above the lowest level, the sequence will be a subset of the sequence at the bottom level. Certain points in time may be covered more than once at least two reasons, (i) if external demand includes events at distances equal to combinations of lead times, and (ii) if combinations of lead times happen to be equal (such as if

τ

₂ were equal to

τ

₃ above).

Figure 3.2. Cumulative requirements as a staircase.

In total, we can therefore regard cumulative requirements on any level (including the top level) to be made up of a staircase of steps occurring at the times t₀, , t₁ t₂, … , of which several steps may have a zero height.

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These requirements are given by

(

)

ˆ ( )s = ( ) ( )s s + /

D Hτ P D s , (3.12)

for a general production policy, and in the general L4L case by (see Section 3.2.1 below)

(

)

1

ˆ ( )s = − ( )s − /

D I Hτ D s (3.13)

When there is only one end item in the L4L case, we have

(

)

1 1 1 ˆ ( ) ( ) / col s =⎡_⎣ − s − ⎤_⎦ D D I Hτ s (3.14)

The staircase is illustrated in Figure 3.2 for an individual item.

3.2 Time Development of Production for Basic

Ordering Policies

3.2.1 Lot-For-Lot (L4L) policy

Figure 3.3. The Lot-for-Lot Policy. Production of an item (thick staircase) follows exactly total requirements (sum of independent and

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In the L4L case, with zero initial available inventory R(0) = 0, the solution is perfectly simple, also in the multi-item case with non-zero lead times. It is then just a matter of keeping available inventory at a zero level. This means

(

)

(0) ( ) ( ) ( ) ( )s s s s s + − − =R I Hτ P D R _{= 0}

_{, (3.15)}

from which production is determined as

(

)

1 _ˆ

( )s = − ( )s − ( )s = ( )

P I Hτ D D s , (3.16)

as is illustrated in Figure 3.3. If there are non-zero initial available inventories, the expression needs to be adjusted slightly.

3.2.2 Fixed Order Quantity (FOQ) case

A fixed order quantity policy may be specified for any item in an MRP system. However, in practice it would be applicable to items with ordering cost sufficiently high to rule out ordering in net requirement quantities, period by period. The replenishments occur as available inventory approaches to zero.

With the FOQ policy, having Q as the fixed order size, production for an individual item will behave according to

ˆ 1 0 ( ) n n sT n P s Q e − − = =

∑

_{, (3.17)}

where is the time when the nth batch is completed (made available), see Figure 3.4. These are the variables that need be determined by the policy. When production is other than L4L, the requirements on lower levels will depend on production on levels above. This implies, in the general case, that new sets of possible times of requirement events might be introduced.

n T

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Figure 3.4. The Fixed Order Quantity Policy (FOQ). Production of an item is made in equally-sized batches as late as possible when avoiding

negative available inventory.

Available inventory of this individual item will then be

ˆ 1 1 0 ˆ ( ) (0) n ( ) n sT n R s R Q e D s s − − − = ⎛ =_⎜ + − ⎝

∑

⎠ ⎞_⎟ _{, (3.18)}

and the policy is to make all as late as possible without causing R(t) to become negative.

n

T

3.2.3 Fixed Period Requirements (FPR) case

Under the FPR policy, the ordering interval is constant and the quantities are allowed to vary. Production of a certain item will thus obey

ˆ 1 0 ( ) n snT n n P s Q e − − = =

∑

_(3.19)

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where T denotes the constant time interval between orders and is the batch size at time

n Q n

T =nT. The FPR policy is demonstrated in Figure 3.5.

Figure 3.5. The Fixed Period Requirements Policy (FPR). Production of an item is made in as small batches as possible to cover future requirements during a constant period, without creating a negative

available inventory.

The FPR policy requires the to be made as small as possible without violating the non-negativity condition for available inventory. This implies that available inventory is likely to take on a zero value during finite time intervals, which is not the case, other than by chance (or at the end), with the FOQ policy, as seen when comparing Figures 3.4 and 3.5.

n Q

3.3 Solutions to Non-Negativity Conditions for

Available Inventory with Requirements as Discrete

Events

As shown above, the L4L policy provides an immediate explicit expression for the production on all levels. For the other two policies, this is not equally simple.

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In the FOQ case we need to solve for the latest as possible batch times , , , … , such that available inventory R(t) is kept non-negative. The solution in the time domain is simple for the individual item, given the requirements

0

T T1 T2

ˆ ( )t_i

D at the times when there are steps t₀, , t₁ t₂,… Examining, for successive values of n,

(

ˆ 1 ˆ

)

arg max ( ) ( ) 0

i

i i

t

D t₋ ≤nQ<D t ≥ , there will be a unique index i assigned to each n, which we denote

i

_n. So,

n n i

T = will be the latest t time that batch n can be produced. Hence, the solution to the FOQ production staircase is:

ˆ 1 0 ( ) in / n sT n P s Q e s − − = =

∑

_{. (3.20)}

In the FPR case instead, we need to solve for the smallest possible batch size at times 0, T, 2T, … , not violating the non-negativity of available inventory. The sequence of batches Q_n generated by

1 0 ˆ (( 1) ) n n i Q D n T Q − = = + −

∑

_i, (3.21) or, equivalently, ˆ₍₍ _{1) )} ˆ₍ ₎ n Q =D n+ T −D nT , (3.22)

for successive values of n = 0, 1, 2, … , will uniquely determine the production staircase satisfying the conditions. Hence, the production staircase becomes:

(

)

ˆ 1 0 ˆ ˆ ( ) (( 1) ) ( ) / n snT n P s D n T D nT e s − − = =

∑

+ − . (3.23)

Until now, we have investigated consequences of the non-negativity requirements in the time domain.

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A corresponding set of non-negativity conditions in the frequency domain is given in Feller (1971) as the following provisions. If a time function ( )f t having the transform ( )f s is non-negative in the time domain, then the following property must hold

( )

( ) 1 1 j j j j j d f s f s ds − = − ( ) 0≥ 1 − , (3.24)

for all integers j > 0 and all real s.

In our case, we are looking at available inventories which for the FOQ policy are given by (assuming initial available inventory to be zero)

{ }

ˆ 1 1 1 0 ˆ ( ) £ ( ) £ n ( ) n sT n R t R s Q e D s s − − − − = ⎧⎛ ⎞ ⎫ = = _⎨_⎜ − _⎟ _⎬= ⎝ ⎠ ⎩

∑

⎭

(

)

ˆ 1 1 1 1 0 1 ˆ ˆ £ n ( ) ( ) n sT st l l n l Q e D t D t e s − − − − − − = = ⎧⎛ _⎛ _⎞ ⎫ = _⎨_⎜ − _⎜ − _⎟ _⎬ ⎝ ⎠ ⎝ ⎠ ⎩

∑

⎭ 0 l ⎞ ≥ ⎟ (3.25)

Choosing the situation for the interval of the ith requirements step and mth batch, and writing

(

)

1 1 0 1 ˆ ˆ ( ) n ( ) ( ) m i sT st l l n l f s Q e D t D t e − − − = = ⎛ = − _⎜ − ⎝ ⎠

∑

− l⎞⎟ , (3.26) 1 ( ) g s =s− , (3.27)

we apply Euler’s formula

(

)

( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) j j k j k j f s g s f s g s k − = ⎛ ⎞ = _{⎜ ⎟} ⎝ ⎠

∑

k (3.28)

to Eq. (3.26). Differentiating ( )f s and ( )g s the number of times required, we obtain

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( )

(

)

1 ( ) 1 0 1 ˆ ˆ ( ) n ( ) ( ) l m i k _sT k _s k n l l l n l t f s Q T e t D t D t e − − − − = = =

∑

− −

∑

− − _{, (3.29)} 1 ( )j k ( 1)j k( )! g s − = − − j−k s−− +j k, (3.30) and thus

( )

(

)

( )

( ) ( ) 0 ! 1 ( ) ( ) 1 ( ) ( ) !( )! j j j j _k _{j k} k j f s g s f s g s k j k − = − = − −

∑

₌

( )

₍

₎

( )

1 1 1 0 0 1 0 ! _ˆ _ˆ ( ) ( ) ! ! n l k k j j m i n l sT st l l j n k l k sT st j Q e D t D t e s k − − − − + = = = = ⎛ ⎞ ⎜ ⎟ = − − ⎜ ⎟ ⎝

∑

k ⎠

(

)

1 ! _{ˆ ( )} 0 l j j mQ D t s + → − ≥ , (3.31)

for large values of j. This again provides the result in (3.20).

3.4 Solutions to Non-Negativity Conditions for

Available Inventory with Requirements as

Continuous Events

In cases when the cumulative requirements are assumed to be a continuous time function given by an analytical expression, we may apply Cauchy’s Residue Theorem for solving for production in the FOQ and FPR policy cases. A residue is the coefficient of the first negative power in a Laurent expansion around a pole, i. e. where the numerator of an expression evaluates to zero.

For the FOQ policy, we need to solve for the points in time when available inventory drops to zero.

{ }

₁ ˆ 1 1 1 0 ˆ ( ) £ ( ) £ n ( ) n sT n R t R s Q e D s s − − − ₋ ₋ = ⎧⎛ ⎫ = = _⎨_⎜ − ⎞ _⎬ ⎝ ⎠ ⎩

∑

⎭ _⎟ _{. (3.32)}

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Assuming zero initial inventories and writing the Laurent expansion of as

ˆ ( )

D s

ˆ ( ) j j j D s d s ∞ =−∞ =

∑

(3.33)

for the nth batch, we have

1 _ˆ

( )

/

( )

2

i _wt w i

R t

nQ w D w

i

β β

π

+ ∞ = − ∞

⎛

⎞

=

_⎜

−

_⎟

⎝

⎠

∫

e dw

, (3.34)

which evaluated by the Residue Theorem will be

1 _ˆ ( ) / ( ) 2 i _wt w i R t nQ w D w e i β β π + ∞ = − ∞ ⎛ ⎞ = _⎜ − _⎟ ⎝ ⎠

∫

dw= 0 ˆ Res ( ) wt w residues nQ D w e w = ⎛ ⎞ ⎛ ⎞ = _⎜ _⎟− _⎜ _⎟ ⎝ ⎠

∑

⎝ ⎠ = 0 ( ) 0 ! k j j residues j k wt nQ d w k ∞ ∞ =−∞ = ⎛ ⎞ = − _⎜ = ⎝ ⎠

∑ ∑

∑

⎟ . (3.35)

As an example, when requirements increase linearly and cumulative requirements therefore increase quadratically, cumulative requirements behave according to _{D s}_{ˆ ( )} =_as−3_,

where a is the slope of the linearly increasing requirements. Then, the Laurent expansion collapses into

3 j j j d w aw ∞ − =−∞ =

∑

, (3.36)

with the only non-zero coefficient

d

₋₃

=

a

. In this case, the only pole is at w = 0, so

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3 0 0 0 0 ( ) ( ) Res Res / 2! ! ! k k j j w w j k k wt wt d w aw at k k ∞ ∞ ∞ − = _=−∞ ₌ = ₌ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝

∑

⎠

∑

2 = , (3.37) and 2 ( ) / 2 0 R T =nQ−aT = . (3.38)

The time of the nth batch will be T = 2nQ/a . Hence, cumulative production in the FOQ case will follow

2 /

( ) s nQ a/ n

P s =Q

∑

e− s. (3.39)

In the FPR case instead, production has the structure

ˆ 1 0 ( ) n snT n n P s Q e − − = =

∑

_{, (3.40)}

and available inventory will be

{ }

₁ ˆ 1 1 1 0 ˆ ( ) £ ( ) £ ( ) n snT n n R t R s Q e D s s − − − ₋ = − ⎧⎛ ⎫ = = _⎨_⎜ − ⎞ _⎬ ⎝ ⎠ ⎩

∑

⎭ _⎟ _{. (3.41)}

At the end of the nth step of the production staircase (at t = nT), we have

1 0 1 _ˆ ( ) / ( ) 2 n i _wnT m w i m R nT Q w D w e dw i β β π − + ∞ = − ∞ = ⎛ ⎞ = _⎜ − _⎟ ⎝

∑

⎠

∫

, (3.42)

which evaluated by the Residue Theorem will be

1 0 1 _ˆ ( ) / ( ) 2 n i _wnT m w i m R nT Q w D w e dw i β β π − + ∞ = − ∞ = ⎛ ⎞ = _⎜ − _⎟ ⎝

∑

⎠

∫

= 1 0 0 ˆ Res / ( ) n wnT m w m residues Q w D w e − = ₌ ⎛ ⎞ ⎛ ⎞ = _⎜ _⎟− _⎜ _⎟ ⎝ ⎠ ⎝

∑

⎠

∑

=

PROFIL 23

Thi Thu Thuy Huynh

CAPACITY CONSTRAINTS IN MULTI-STAGE

PRODUCTION-INVENTORY SYSTEMS

– APPLYING MATERIAL REQUIREMENTS

PLANNING THEORY

PROFIL 23

PRODUCTION-ECONOMIC RESEARCH IN LINKÖPING

LINKÖPING 2006

PREFACE

FOREWORD

CONTENTS

1 INTRODUCTION

1.1 Background and Motives

Figure 1.1. An overview of how the MRP system works.

Adopted from Foster (2002).

Combining

Laplace transforms

1.2 Objectives

1.3 Limitations

1.4 Outline and Scientific Contribution

2

LITERATURE REVIEW AND RESEARCH

FRAMEWORK

2.1

Capacity-Constrained Multi-Stage Production

Planning and Inventory Control

2.2 The Application of MRP Theory Employing

Laplace Transform and Input-Output Analysis to

Multi-Level Multi-Stage Production and Inventory

Systems

2.3 Capacity-Constrained Multi-Level Multi-Stage

Production Inventory Systems – Some Notation and

Explanations.

∫

{ }

{

}

∫

=

∑

∫

∑

(

)

(

(

)

(

)

(

)

)

, (2.1)

w)

( )

(

)

(

)

(

)

[ ]

( )

s

P

{ }

{ }

3

ANALYSIS OF STANDARD ORDERING

POLICIES WITHIN THE FRAMEWORK

OF MRP THEORY

3.1 The

Fundamental

Equations of MRP Theory

(

)

(

)

(

_{( ) ( )}

_{, (3.15)}