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#### Management of multi-echelon inventory systems: Methodological proposal for decision support using the system dynamics approach and multicriteria analysis

Intellectual Capital, Management Theory March 5, 1996, 0 Comment 237**Abstract:**

This paper presents a methodology for analyzing management problems using both the Systems Dynamics Approach and the Multicriteria Decision Aid Methodology (MMCAD). This methodology, called Interactive Decision Support Methodology (MIAD), is demonstrated through the study of multi-echelon inventory system.

The MIAD is composed of three phases: Problem definition, project planning and project execution. This methodology intends to use the System Dynamics Approach as a tool to:

- facilitate the process of problem emergence and definition,
- identify possible alternatives,
- evaluate the consequences of implementation,
- make explicit the consequences of the dynamic behavior of the system.

The MMCAD is used as a tool to find a coherent criteria family and a preference system over the alternatives.

The example problem is to find the best management policy for an inventory system that could be implemented as a one or two-echelon system. The alternatives are the different permutations of the two most used inventory policies: Periodic Review and Reorder point. The result is that the reorder point policy with a single echelon inventory systems is the best.

The MIAD proved to be very useful to show the relevant characteristics of the systems behavior by using dynamic simulation and to achieve a concise prescription by using the Multicriteria Decision Aid Methodology (MMCAD).

**I.****– The Interactive Decision Support Methodology.**

**1.- Introduction.**

The purpose of this paper is to demonstrate that the joint use of the System Dynamics Approach (WOLSTENHOLME 1990) and the Multicriteria Decision Aid Methodology of B. ROY (MMCAD) (BOUYSSOU 1993, ROY 1985) facilitates as well as increases the effectiveness of the decision making process in the organization.

The paper is divided in three parts: first the principles of the methodology, second the application of the methodology to the problem of inventory management and last the overall conclusions and recommendations.

Among the different support tools for the solution of management problem one of the most effective tools is the System Dynamics Approach because it facilitates the understanding of the relationship between the behavior of a system over time and its underlying structure and strategies/policies/decision rules (WOLSTENHOLME 1990 p. 2, MORECROFT 1990). Despite this advantage, its application to practical problems has demonstrated the requirements of a tool for the analysis of the outcomes. This was the origin of the use of the multicriteria analysis as an evaluation tool.

One of the most effective approaches in Decision Science is the Multicriteria Methodology for Decision Aid (MMCAD) (BOUYSSOU 1993, ROY 1985), basically because it gives tools for the management of uncertainty and imprecision as well as handling adequately the detailed level of information that usually is found in practical studies. It also provides tools for the definition and management of several criteria, allowing the exploration of the value system of the actors present in a decision process in order to arrive to a recommendation that fits into the process requirements. One weakness of the multicriteria decision support systems is the need for tools that simplify the problem emergence process (POMEROL 1993). For this reason it is interesting the complementation the MMCAD with a System Dynamics approach.

The multi-echelon inventory management problem was chosen because it has been already studied employing the MMCAD, which permits the validation of the output of the present study (VALBON-SUTOUR 1993).

**2.- Hypothesis.**

To propose a methodology of management decision aid a set of hypothesis must be defined. The hypothesis of the MIAD are the following:

An organization is a complex system whose behavior depends on the interaction of its elements and thereby of its structure (ARACIL 1984).

The elements that make up an organization are its actors. They are oriented towards particular objectives that trigger the decision making .

The behavior of actors in a management situation[1] is based on routines (MARCH 1991) and the behavior over time of an organization is a consequence of the interactions between the routines.

These routines can be made explicit through cognitive maps (COSSETTE 1988). The hypothesis is that the routines of the actors are a consequence of their mental models, which can be represented using the causal relations established by the individual. A cognitive map is a graphic representation of the causal structure of a system.

Once the representation of the decision routines of a set of actors is given, the tendencies of the system can be assumed.

Through the setting of adequate evaluation parameters, criteria, a preference system can be established that allow the decision makers to take coherent decisions within their value system.

**3.- Methodology.**

**3.1.- Overview.**

Based on the previous hypothesis set, a work methodology has been proposed that has three stages:

- Problem formulation.
- Project planning.
- Project execution

The key point in the solution of a problem is its formulation. In many cases the correct formulation of a problem leads directly to its solution, so it is proposed that exhaustive research be conducted for the definition of the problem. If an obvious solution is not found through the formulation, then the continuation of the study is decided.

To use a particular methodology or technique for the resolution of a problem implies taking as truth a set of hypothesis that will affect the solution. This is why there is a stage dealing with the analysis to determine the technique to be used and planning the steps that will be followed in the study.

Once decided and planned, the study must be performed. For more details on the given methodology see ROSAS FLUNGER (1993).

**3.2.- Problem formulation stage.**

The problem formulation stage involves two basic views, the conceptualization idea, from the System Dynamics approach (RANDERS 1980) and the ideas of prescription and participation as defined in level I of the MMCAD (ROY 1985).

The first step in the methodology is to make explicit the assumptions of the problem. Therefore a representation language for the problem must be defined. The Causal Diagrams and Influence Diagrams[2] (COSSETTE 1988, MERKHOFER 1990) are visual tools that have proven to facilitate the recognition and definition process of the causal relations in a problem (HODGSON 1992, ARACIL 1984, RANDERS 1980). Since the first step of formulation is to make explicit the problem hypothesis, the MIAD proposes the elaboration of:

i.- Causal Diagram of the System. Which implies the identification of the different elements of the system and its interrelations (COSSETTE 1988).

ii.- Influence Diagram of the Objectives. Which include the different elements which influence the objectives of the decision maker (MERKHOFER 1990).

Once the Diagrams are ready the analysis follows and if an evident solution is not found, a system dynamics simulation model is built. Hence, the adequate modelling technique should be used (WOLSTENHOLME 1990):

i.- Feedback loop approach.

ii.- Modular Approach.

The feedback loop approach consists on the identification of the feedback structure by matching system behaviors with feedback loops already known. These behaviors are the positive and negative feedback loops of first and second order. This identification is necessary to have a previous behavior as reference, that is, a history or a specific reference to the behavior of the system.

The modular approach consists on the identification of all the elements that play a part in the system behavior and progressively find the interrelation links between them. Consequently, different tools of Causal Diagrams and Influence Diagrams and systems dynamics modelling will be used (COSSETTE 1988, MERKHOFER 1990, WOLSTENHOLME 1990, RANDRES 1980, HODGSON 1992).

Based on the simulation modelling methodology (LAW 1991) and on the fact that they are decision support models, the following elements must be described:

i.- Objectives.

ii.- Actors.

iii.- States.

iv.- Events.

a.- Physical events.

b.- Decisions.

v.- Routines.

a.- Physical Routines.

b.- Decision Routines.

vi.- Decisions Variables.

vii.- Control Variables.

viii.- Statistical Counters.

Once assembled, the simulation model must be validated following several steps. The structure of the model must be validated, which implies the verification of its concordance with the ideas of the decision maker, the validity of the parameters, the adequately of the boundaries and the behavior of the system, (WOLSTENHOLME 1990 p. 60). The tests that must be accomplished are:

1.- Model structure validation.

i.- Cognitive structure verification.

ii.- Parameter verification.

iii.- Extreme conditions

iv.- Boundary adequacy.

v.- Dimensional consistency.

2.- Model behavior validation.

i.- Behavior reproduction.

ii.- Behavior robust analysis.

iii.- Matching with known behavior.

Once the model is accepted as valid, its analysis must be performed. The purpose is the identification of possible actions[3]. Hence, it is proposed the methodology analysis of WOLSTENHOLME (1990 pp. 29-31) which consists of the following steps:

1.- Isolate the major feedback loops in the model.

2.- Identify the general mode of behavior of the individual loops and the whole system.

3.- Identify the variables within each loop which can be controlled.

4.- Identify possible ways to control these variables.

5.- Reiterate if necessary.

Once the system is analyzed, the actions and obvious solutions (policy) must be identified and the problem solved . If an evident solution is not found a more detailed study must be accomplished. Once decided that a study will be made, the sense of prescription and the final problem structure must be defined.

The Problem Definition stage can be resumed in the following steps:

**Phase I****: Definition of Hypothesis and Diagrams.**

1.- Hypothesis formulation.

2.- Causal Diagram of the system.

3.- Influence Diagram of the objectives.

4.- Diagram Analysis.

**Phase II****: Modelling.**

1.- Feedback loop approach.

2.- Modular approach.

3.- Elements:

i.- Objectives.

ii.- Actors.

iii.- States.

iv.- Events.

a.- Physical events.

b.- Decisions.

v.- Routines.

a.- Physical Routines.

b.- Decision Routines.

vi.- Decision variables.

vii.- Control variables.

viii.- Statistics counters.

**Phase III****: Validation.**

1.- Validation of the model structure.

i.- Verification of the cognitive structure.

ii.- Verification of parameters.

iii.- Extreme conditions

iv.- Boundary adequacy.

v.- Dimensional consistency.

2.- Validation of the model behavior.

i.- Behavior reproduction.

ii.- Robust analysis of the behavior .

iii.- Matching with known behavior.

**Phase IV****: Analysis.**

1.- Isolate the major feedback loops in the model.

2.- Identify the general mode of behavior of the individual loops and the whole system.

3.- Identify the variables within each loop which can be controlled.

4.- Identify possible ways to control these variables.

5.- Reiterate if necessary.

**Phase V****: Decision.**

1.- Identification of actions and evident solutions. End of the project.

2.- Decision to continue the project.

3.- Final definition of prescription sense and problem structure.

**3.3.- Project planning stage.**

The project planning means the definition of the technique to be employed. Since every technique has its own, methodology the selection of a specific one determines the subsequent steps of the study. The MIAD only explores the interaction between System Dynamics and the MMCAD. This fact does not mean that they are the only techniques available.

First, it must be defined whether the project justifies the use of dynamic simulation. If the evolution over time of the system behavior proves to be important, the dynamic simulation must be done. This decision is a consequence of the outcomes of the system behavior tests. If it is decided to continue through the dynamic simulation the generic actions would be defined, that means, the parameters and general characteristics of the potential actions and research directions would be defined. In conclusion this stage will include:

1.- Definition of the analysis technique.

2.- Definition of the generic actions and research directions.

**3.4.- Project execution stage.**

The initial step in the execution of the project using system dynamic simulation is the exact definition of the actions and the problematic reference[4] of the study.

The definition of the actions follows a divergent thinking process, not accurately structured, hence it is strongly supported by intuition and experience.

Level II of the MMCAD (ROY 1985) follows and the simulation parameters are selected. The simulations and statistical analysis are performed and with the outcome the comparative analysis is done (level II and III of the MMCAD, ROY 1985). Thus, this stage is composed of the following steps:

**Phase I****: Experience design.**

1.- Definition of the actions.

2.- Definition of the problematic.

3.- Consequence analysis and criteria construction.

i.- Definition of the scales and dimensions.

ii.- Definition of the method for action evaluation.

iii.- Definition of the criteria and criteria functions.

iv.- Definition of the coherent criteria family.

4.- Definition of the simulation parameters.

i.- Simulation time span.

ii.- Number of replications.

iii.- Variance reduction technique.

iv.- Output reports.

**Phase II****: Model exploitation**

1.- Simulation runs.

2.- Statistical analysis of simulation output.

3.- Sensitivity analysis.

4.- Simulation documentation.

**Phase III****: Comparative analysis.**

1.- Modelling of global preference system.

2.- Definition of the operational approach.

3.- Definition of the research procedure.

4.- Prescription formulation.

**3.5.- Methodological proposal.**

The methodology could be summarized as follows:

I.- Problem formulation.Phase I: Hypothesis and Diagram definition.Phase II: Modelling.Phase III: Validation.
Phase IV: Analysis. Phase V: Decision. |
II.- Project planning.Phase I: Analysis technique definition.Phase II: Generic action and research directions definition. |
III.- Project execution.Phase I: Experience Design.Phase II: Model Exploitation.Phase III: Comparative analysis. |

**II.- Example: Multi-echelon inventory management**

**1.- Generalities.**

The problem to be studied is the selection of an optimal policy of inventory management. For this problem the propositions of M-C VALBON-SUTOUR (1993) were followed. The basic idea is to compare inventory management policies with multiple criteria and to make the cost evaluation a posteriori. One of the most studied problems in inventory optimization is the cost determination. To measure, in monetary terms, the impact of inventory policies is not simple. To transform quality and efficiency parameters into economic values is complex. That is one of the reason to take VALBON-SUTOUR ideas of doing a multicriteria analysis, identifying the parameters that determine a correct inventory policy and evaluate its cost a posteriori. The comparisons are made between reorder point and periodic review policy.

The study compares the cases of one or two level inventory systems. The first case is a warehouse that satisfies directly the client’s demand and it is furnished by a supplier. The second case is when a central warehouse supplies the demands of local warehouses that in turn satisfy the client’s demand. The central warehouse is the only one who is furnished directly by the supplier. These systems are called “multi-echelon” in the literature (see GIARD 1981 p. 139).

The periodic review and reorder point policies implementation means, in the multi-echelon case, to define the way of taking into account the inventories that are distributed in the different warehouses. Hence, two options are defined: restrained option and extended option. The restrained option is characterized by only taking into account the central warehouse inventory level for the calculation of the replenishment variables. The case when the central and local warehouse inventory level are taken jointly into account for calculations is the extended option.

Various ways to use the generic policies are possible for the two level systems. The pure policy is the one that uses the same policy for the central and local warehouses although they can use different parameters. The mixed policy is the one that uses the two generic policies, that means, the central warehouse one and the local warehouses the other. It is important to realize that is possible the case where the local warehouses use both policies, that means, some with periodic review and others with reorder point.

A multi-echelon inventory study has to consider the following four conditions:

i.- If the system demand is completely satisfied by the supplier.

ii.- If a known delay exists between the central and local warehouses.

iii.- If a known delay exists between the local warehouses and the clients.

iv.- If the unsatisfied demand accumulates or is lost.

**2.- Problem formulation stage.**

**2.1.- Phase I: Definition Hypothesis and diagrams.**

**2.1.1.- Hypothesis formulation.**

The problem is to find the inventory system and its control policy which can best satisfy a random demand that follows a normal distribution with a mean and variance known and which could be desegregated in to five independent demands. The alternatives are the different inventory systems, one level or two levels, the different policies, reorder point or periodic review, the different combinations, pure or mixed, and the different options, restrained or extended.

The functioning hypothesis of the system are:

i.- The system demand is satisfied completely by the supplier.

ii.- A known delay between the central warehouse and the local warehouses exists.

iii.- A known delay between the local warehouses and the clients exists.

iv.-The unsatisfied demand is lost.

For more details on the operation of the system see ROSAS FLUNGER (1993) and VALBON-SUTOUR (1993).

**2.1.2.- Causal diagram of the system.**

The causal diagram of the system was drawn using the FORRESTER diagram symbology (ARACIL 1984, WOLSTENHOLME 1990) with a simulation shell called iTHINK v. 2.01 (RICHMONDS 1991).

The first representation that must be accomplished in a system is its physical behavior, in this case, the placing of orders, the arrival of products and the consumption of the local warehouses and clients (see Fig. 1).

**Fig. 1**: Physical transit diagram.

The inventories in the warehouses are represented as levels[5]. The orders from clients and from warehouses are represented as flows. The product transit occurs at physical channels. The other variables are taken as auxiliary variables.

**2.1.3.- Influence diagram of the objectives.**

The objective is to determine the best inventory policy. In building such an influence diagram it is assumed that these objectives depend on two basic factors: cost minimization and client satisfaction maximization (see Fig. 2).

**Fig. 2**: Influence diagram of objectives.

Three factors are the determinant elements in the inventory cost: financial cost, storage cost and ordering cost. These costs depend on inventory volume and the number of orders.

The maximization of client satisfaction depends on the frequency of out of stock runs.

**2.1.4.- Diagram analysis**

Analyzing the diagrams it is evident that in order to make a decision it is important to determine the inventory behavior which depends on the delivery and client demands, consequently the factors that affect these elements must be analyzed in detail.

Other important point is to define the interrelation between the objectives and the system parameters. In conclusion a more detailed modelling stage must be accomplished.

Another conclusion is that the significant elements that affect the general objective of the problem are the inventory level, the number of orders and the frequency of out of stock runs.

**2.2.- Phase II: Modelling.**

i.- Objectives: To determine the inventory levels, the number of orders and the frequency of out of stock runs of the different inventory management systems.

ii.- Actors: The warehouses, supplier and clients.

iii.- States:

– Inventory level in the central and local warehouses.

– Back-Orders.

– Total number of out of stock runs.

– Total orders.

iv.- Events:

a.- Physical events.

– Products arriving to warehouses.

– Product delivery to warehouses and clients.

b.- Decisions.

– Warehouse orders.

– Client and warehouse delivery.

v.- Routines:

a.- Physical routines.

– Process of products arriving to warehouses.

– Process of product delivery to warehouses and clients.

b.- Decisions routines.

– Ordering.

– Delivery

vi.- Decision variables:

– Order periodicity.

– Protection levels of warehouses.

vii.- Control variables:

– Mean and standard deviation of client consumptions.

– Initial inventory of warehouses.

– Mean and standard deviation of delivery delay of warehouses.

– Seeds of random variable generators.

viii.- Statistical counters.

– Individual and global frequency of out of stock runs.

– Individual and global inventory levels.

**2.3.- Phase III: Validation.**

Although the model structure was found to be valid, behavior problems were found. A model behavior is considered valid, when in the deterministic domain completely satisfies the demand.

The system behavior did not match the theory due to several reasons. First there was a problem with the precedence order that the simulation shell was using to execute the operations. Second, in the simulation runs, the shell added an additional delay of one time unit on the delivery which resulted in the need to revise of all operating formulas.

**Fig. 3**: Graphic of sensitivity runs over quality.

The formulas to determine the central warehouse security threshold for the mixed policy extended option did not give the adequate value to satisfy the clients demand. Different essays were done in the deterministic domain adding a coefficient that varied from zero to two. The impact in the service quality can be seen in Fig.3.

**2.4.- Phase IV: Analysis.**

Analyzing the outcomes of various test runs it was found that a significant number of parameters needed to be accurately determined to define the actions and possible solutions.

It was also found that the system behavior presented deviations due to a convergence phenomenon in the system parameters.

**2.5.- Phase V: Decision.**

Since the first analysis of the problem did not conclude with a solution it was decided to continue the study. The prescription sense is to recommend a management policy for the stock system.

**3.- Project planning stage.**

**3.1.- Phase I: Definition of the analysis technique.**

From the problem formulation, and analyzing different propositions existing in the literature (AXSASTER 1990, ROGERS 1991) it was decided not to look for an analytical solution but to use the simulation technique (LAW 1991).

The random behavior of the demand determined the use of stochastic simulation and the variable behavior over time determined the use of dynamic simulation.

Analyzing the problem objectives it is clear that several criteria must be optimized which suggest the use of multicriteria analysis.

In conclusion, the proposition was to use jointly dynamic simulation and multicriteria analysis (DORF 1991).

**3.2.- Phase II: Definition of the generic actions and research directions.**

i.- Generic actions.

The generic actions to be investigated are the different inventory systems and their corresponding management policies. Physically two options exist: one level and two levels.

The management policy alternatives are two: periodic review and reorder point. This implies that for the two levels case there are various alternatives. One is that the system works with only one policy, defined as pure policy. The other alternative is that the system works with mixed policies which means that the central warehouse uses one policy and the local warehouses use the other policy.

There is the option that the local warehouses have different policies among them. For the case study, all local warehouses used the same policy.

Another possibility exists, defined by the way of accounting the inventory levels: restrained or extended.

The generic actions are the different inventory management systems resulting from the permutations of this set of options. They are the following:

1.- One level, periodic review (GP1N).

2.- One level, reorder point (GPC1N).

3.- Two levels, periodic review pure, extended option (GP2NPE).

4.- Two levels, periodic review pure, restrained option (GP2NPR).

5.- Two levels, reorder point pure, extended option (GPC2NPE).

6.- Two levels, reorder point pure, restrained option (GPC2NPR).

7.- Two levels, periodic review mixed, extended option (GP2NME).

8.- Two levels, periodic review mixed, restrained option (GP2NMR).

9.- Two levels, reorder point mixed, extended option (GPC2NME).

10.- Two levels, reorder point mixed, restrained option (GPC2NMR).

The nomenclature used is that the two-level inventory systems that use a mixed policy are named by the central warehouse policy. For example, the *two levels, reorder point, mixed, extended option*, consists in a two levels inventory system that uses reorder point policy for the central warehouse and periodic review option for the local warehouses and estimates the level of the whole system for the security threshold.

ii.- Research directions.

First it was necessary to define in detail the formulas and parameters of the ten actions.

To compare one and two level systems they must be effectively comparable. Hence, the one level client demand is the aggregation of the client demand of the two levels systems local warehouses (VALBON-SUTOUR 1993 p. 403).

The comparison must be done in steady state (LAW 1991), thus the parameters that stabilize the systems must be found.

The sensitivity analysis must study:

a.- System stability.

b.- Criteria sensitivity to variations in the decision variables..

c.- Criteria sensitivity to delivery delays and standard deviation of the demand.

The system stability depends on the initial conditions, the interrelations between the central and local warehouse periodicity of order placing and the formula parameters. Therefore these three factors will be studied.

**4.- Project execution stage.**

**4.1.- Phase I: Experience design.**

**4.1.1.- Definition of actions.**

The exact definition of the actions means the determination of the periodicity of central and local warehouses order placing and their security level in each generic action.

The use of different periodicities in central and local warehouses causes convergence problems, hence it was decided to use the same periodicity for all warehouses. To find the optimum value of the periodicity involves a specific and complex optimization problem, therefore an apriori value was chosen.

It has been shown that the minimum total expected penalty cost is achieved by allocating to each of the local warehouses a quantity such that the probability of running out of stock is the same for all warehouses (ROGERS 1991).

The actions were defined with a three week order periodicity and a security level of ninety five per cent (95%). For the calculation, a normal distribution law was used to transform the security level.

The initial inventories were defined according to the particular management policy of each action in order to guarantee the stability. For more details in formulas see ROSAS FLUNGER (1993) and VALBON-SUTOUR (1993).

**4.1.2.- Problematic definition.**

The problematic is to choose (P. a) the best inventory system and its respective management policy (ROY 1985 pp. 74-95).

**4.1.3.- Analysis of the consequences and criteria construction**

i.- Definition of scales and dimensions: The system behavior is revealed by the order periodicity, the inventory level and the number of weeks that a warehouse of the system runs out of stock. A warehouse runs out of stock when it is not able to satisfy the client orders.

The scales are:

– weeks between orders.

– quantity of product in inventories.

– number of weeks that run out of stock.

ii.- Definition of the action evaluation method: The action evaluation is done following the evolution of the number of orders made by the system, the inventory level and the number of times that the system runs out of stock in a time interval. The evaluation was effected at one and two years.

iii.- Definition of criteria and criteria functions: The criteria to be taken are the ones defined by VALBON-SUTOUR (1993):

a.- **Order periodicity:** Defined as the mean number of weeks between two sequential orders to the supplier. This criteria must be maximized.

b.- **Service quality: **Percentage of weeks that the system do not run out of stock in a time interval. This criteria must be maximized.

c.- **Inventory level : **Average number of weeks of demand in inventories. This criteria must be minimized.

The criteria functions are in ROSAS FLUNGER (1993) pp. 73-74.

iv.- Definition of the coherent criteria family: The coherent criteria family corresponds to the three preceding criteria.

**4.1.4.- Definition of simulation parameters.**

i.- Simulation time span: The time of each simulation is 104 weeks to allow the comparison at 52 and 104 weeks (one and two years). The simulation step size is one week (dt=1 week).

ii.- Number of replications: In order to have statistical validity in the output five replications of 104 weeks were made.

iii.- Variance reduction techniques: The variance reduction technique used was the common random variates. Hence, common seeds in the random variates generator of the simulation shell were used.

iv.- Output reports: Two kind of reports were prepared, numerical and graphical. The numerical reports consist of tables with the simulation outputs and the graphical reports are comparative curves with the system variables (ROSAS FLUNGER 1993). Fig.4 shows the graphic of the criteria evolution over time.

**Fig. 4**: Criteria evaluation graphic.

**4.2.- Phase II: Model Exploitation.**

**4.2.1.- Simulation runs and statistical analysis of the simulation output.**

The criteria values at week 52 and 104 were taken and the means of the five replications of each action were analyzed. The outcomes at weeks 52 and 104 are presented in the following tables (Table Nº 1 & 2).

**4.2.3.- Sensitivity analysis.**

The decision variables have a significant impact on criteria evaluation. The delivery delay of the suppliers and the standard deviation revealed to be of low impact on the criteria evaluation.

Problems were found in the extended option, due to the implicit processing time of the system. These problems were solved modifying the formula coefficients. The simulations used for the final analysis were done with the modified formulas.

The initial conditions, that in the system were the initial inventories, revealed to be determinant in the convergence times of the criteria. This fact agrees with dynamical systems theory.

Particular behaviors were found caused by the numerical relations between the periodicities of the central and local warehouses. These problems are related to the convergence and amplification phenomenon common in dynamical systems in the synchronicity phenomenon of periodic behavior.

**4.3.- Phase III: Comparative analysis.**

**4.3.1.- Global preference system modelling.**

A global preference system of (P,I)[6] type was assembled, eliminating all incomparability. The criteria employed, Periodicity, Quality and Inventory level were taken as true criteria (ROY 1985).

**4.3.2.- Definition of the operational approach.**

The single syntheses criteria approach was chosen based on the following:

– The scales are sufficiently clear and quantitative.

– The scales can be expressed by real units, weeks between orders, percentage of weeks that run out of stock and weeks of the mean demand in inventories.

– Simple manipulations can be made to find a single evaluation scale.

An aggregation function of the three criteria was defined in order to build a true syntheses criteria. Although it is important to make some remarks.

The outcomes have a strong component of randomness due to the fact that they were the simulation output, therefore it is important to make the pertinent remarks to their probability.

The comparison between one level and two level can be questionable and considerations arise about incomparability or the need of other criteria.

**4.3.-3.- Definition of the research procedure.**

The weighted sum was taken as the multicriteria aggregation procedure due to its clarity and ease of implementation.

The outputs were normalized to uniform the different criteria scales. The chosen procedure consists in dividing the studied value by the maximum value of the value set, thus preserving the proportionality and cardinality of the original scales (BARBA-ROMERO 1993 p. 69).

Other important consideration was the transformation of the criteria Inventory level before normalization. Due to the fact that the optimization direction of the other two criteria is maximization and the Inventory level must be minimized, the values were inverted (1/x) to maximize the weighted sum. This was done to preserve the proportionality of the original scale.

The weights were chosen in order to give a bigger importance to quality and the same weight to the periodicity and the inventory level. The weights were normalized dividing the values by the sum of the value set.

The normalization outputs are shown in the following tables:

The criteria aggregation outputs are in Table Nº5 and the final ranking of the actions is in Table Nº6:

**4.3.4.- Prescription formulation.**

The aggregation output obtained after 52 and 104 weeks gave similar rankings. The prescription is to use one level inventory system with reorder point policy.

It is important to mention that the result depends on the weights. If they are modified and more importance is given to the quality, the periodic review policy with one level inventory system becomes the preferred solution.

The standard deviation of the quality criteria in the two levels systems is very high therefore the dispersion must be looked at. Other operational approach or research procedure could be used or other criterias could be taken.

A significant point is that the convergence point depends strongly on the initial conditions, hence attention must be paid to this parameter in the implementation of the solution.

**III.- Conclusions and recommendations.**

From this work, two types of conclusions can be extracted, some from the study outcomes and some from the results of the application of the methodology.

The study outcomes agree with results of other research. Experiences and research on multi-echelon inventory systems have often given the conclusion that the centralized inventories are more adequate than the descentralized to serve a random demand (ROGERS 1991). This result agrees with the prescription of the study. The reflection about re-order point and periodic review policies also agrees with the result of VALBON-SUTOUR (VALBON-SUTOUR 1993).

An interesting field of research is the study of the feedback of the three criteria as decision elements in a system. This field is related to the multicriteria optimization of simulation models, field with not well developed until recently (EVANS 1991).

It can be concluded about the methodology that the use of the Causal Diagrams for the system description allows the identification of the determinant factors in the system behavior. The objectives influence diagram allowed the identification of the determinant elements in the selection of the best system. They are the periodicity, the inventory level and the out of stock runs. The diagram reveals the criteria dependence relations.

The evolution over time of the system behavior was made explicit with the simulation and allowed the model validation as well as to recognize important considerations about the hypothesis set.

Through the simulations, the consequences of actions and criteria can be evaluated.

The MMCAD allowed the coherent analysis of the outcomes to bring about a clear and concise prescription.

The use of MIAD made evident important interrogants related to the inventory system study. The syncronicity links between the periodicity of central and local warehouses, the influence of the initial conditions over the convergence, the validity of the formulas are all future research directions.

An interesting issue is the implementation of interactive automated systems (DSS) that facilitates the use of MIAD . There are several propositions about the use of tools using artificial intelligence for the support of system dynamic models (GOULD 1985).

**References**

**ARACIL, Javier**, (1984), *Introduction à la Dynamique des Systèmes*, Presses Universitaires de Lyon, Lyon, France , 1984.

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[1]: “A management situation appears when the participants are gathered and they have to acomplish a colective action in a given time oriented to a result subject to external judgement” (GIRIN 1990) (Author translation).

[2]: In this paper, the concepts Causal Diagram, Influence Diagram and Cognitive Maps will not be explored. There are authors wich differeciate between them. For more details see ROSAS FLUNGER (1993) pp. 13-14. There will be differenciation if needed.

[3]: In this paper the term “action” is used with the following definition: “An action «a» is the representation of an eventual contribution to the global decision that, in terms of the decision process advancement stage, could be seen in an autonomus way and to serve as application point of the decision aid.” (ROY 1985 pp. 55-71).

[4] : The term problematic is used in the MMCAD sense (ROY 1985 pp. 74-95).

[5]: For details in the System Dynamics vocabulary and symbology see ARACIL (1984), ROSAS FLUNGER (1993) and FORRESTER (1968).

[6] : A (P,I) type preference system consist in two relations of preferences, strict preference “P” and indiference “I”.

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