Engbtecrmg Costs and Production Economics, 7 (I 983) 127 -136

Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands 127

A REPORT OF RESEARCH ON INVENTORY MODELS”

6. Barancsi, G. Bainki, R. Borlbi, A. ChikBn, P. Kelle, T. Kulcs&r and Gy. Meszena

Institute of Mathematics and Computer Science, K. Marx University of Economics,

1093 Budapest, Dimitrov t&r 8 (Hungary)

~- _^-

ABSTRACT

The objective of this paper is to describe the steps in and results of current research carried out within the framework of a special long-term project supervised and financed by ;&he Hungarian Academy of Sciences. Since the research, which started in 1976, will go

on for several more years -- and so we are far from final results - this paper is mostly of an informative nature, intended to make known the ideas behind our work and the progress of our research.

1. BACKGROUND OF THE RESEARCH: INVENTORY MODELS AND THE STATUS OF PRACTICAL APPLICATIONS

It is well known that inventory modelling is one of the most developed fields of opera- tions research. Publications on inventory models could till a small library and in the existing models one can find the most advanced and sophisticated methods of math- ematical theory of operations research. But it is also a fact that the practical implementa- tion of inventory models is no more advanced than the application of other types of opera- tions research models: even attempts at implementation can be found in the case of only very few models, despite the fact that management - at least in principle - needs, these models in practice.

*This article is an extended version of a paper presented at the First International Symposium on Inventories, Budapest, September 1980, which was published in [9].

0167-188X/83/%03.00 0 1983 Elsevier Science Publishers B.V.

The idea to undertake our research came from practice. Members of the research group which we have formed had been involved in modelling for a long time. We learned, created, adapted models and also tried to implement them, and our experience was like those gamed in almost all parts of the world: that is, implementation was hard or impos- sible. There is a demand for models by managers, and there is a supply of models and modelling people, but the two parties just do not meet successfully.

We have analysed the reasons underlying the discrepancy between theory and practice of inventory modeling and found some factors hindering implementation. Namely, in some cases the organizational level of companies does not allow for the creation of a comprehensive model. Sometimes decision- makers are averse to applying models even in cases when the conditions for ‘modelling are given. Other managers would like to apply

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models even when conditions and circum- stances are not appropriate. In a lot of cases computers at companies are used for registra- tion only and are not oriented to decision- making purposes, so they cannot help in im- plementing models. We lnust also state that a great proportion of the models were created because of the mathematical aspect of the problem without even considering the possibility of application.

On the other hand, some factors do help in the application of inventory models. The inventory subsystem of a company can be structured well and modelled relatively easily because it is fairly closed, its input-output connections can be well defined, its objectives can be clearly determined ami inventory deci- sions are mostly periodic routine decisions. Top management is interested in inventcries because they have a very important influence on short-term adaptation of the company. A developed data-processing system which is used for registration can provide a good statistical background for the implementation of models.

By way of conclusion, we believe that while revolutionary change in the near future is unlikely, implementation of inventory models will develop together with the development of organization and data proces- sing of companies and they will play a more important rtile in increasing the efficiency of company operations.

2. OBJECTIVES OF THE RESEARCH

From the above analysis, we have drawn the conclusions that our research should be oriented towards the existing inventory models, in order to a.nalyse them and to build such a computerized model system which is available to managers for practical use. We have set the following objectives in our research: (19 Through the mathematical artd economic

analyses of models, we seek to learn some

aspects of modehing inventories, namely: - what economic (management) situa-

tions are treated in the existing models and what kinds of new models can be or should be buifi CO meet these situa- tions?

- which are the irqortant factors and situations of in9ntoi-y management which do not appear in models, why is that and how CM these factors be in- volved in mode&:

-- what mathematical tools are actually used in the models and how can this range be expanded considering the present stage of mathematical theory?

- what kinds of groupings and generaliza- tions can be given on the basis of existing models?

(2) To promote appfications, we decided to build models into a computerized ir,ventory control system. We wanted to ._ anallyse existing models from the point

of v&v of computer programming; - establish a computerized model library; - work out a model selection procedure,

which makes it possible for managers to get information about potentially appli- caole models without trying to force ihem to learn mathematics.

3. METHODS OF RESEARCH

According to our estimation, there are more than a thousand models in the inter- national specialized literature. We decided to work on an appropriate sample first, and then - if time and our support allows - extend that sample to get as close to the com- plete set as possible,

So far we have collected and studied a sample of 336 models. At the present point, we have stopped working on new models and have started to analyse the sample. The sample can be characterized by the following bibliographical data:

There are IO books (references 1, 2, 4, 5,

1 29

1 O-l 5) among the sources of models (including the classical ones), from which we have taken 40% of the whole sample. About 50% of the sample originates from 12 journals, including Operations Research (40 articles), Management Science (34), Operational Research Quarterly (19) as th,e most important ones. Ten per cent of the models are from research reports, conference proceedings, etc. As for the time distribution, most of the books involved have been pub- lished in the second part of the 1960s. We have balanced the time distribution by including more of the recent issues of the journals. Before 1960 there are seven models; 196165, 66 models; 1966-70, 113 models; 1971-75, 99 models, and 5 1 models from 1976. As for the languages of references: 25 1 English, 38 Russian, 31 German and 16 Hungarian.

3.1 The code system

We have decided to compose a code system, which appropriately describes the characteristics of the models. The code system has been created on the basis of our a priori knowledge of models. Obviously, we had to be extremely careful in selecting those features of the models which will appear in the code system, and also their ways of representation. For over a year we dealt with this problem only, with discussions and experimental model processing. Finally, we have created a code system of 45 codes, each having two to ten possible values. Every code corresponds to a property of models. To describe the whole system would require more than 20 typewritten pages, so we list here only the so-called “main codes”, and give examples to show the structure of the system.

The system of values of these main codes is in itself suitable for the description of the models. These codes represent the main char- acteristics of models, and their values usually have a great influence on the values of other

codes, since the latter actually are to give

more detailed information about the char- acteristics primarily specified by the main codes.

These main codes and their possible values are the following. (1) Number of products:

- one - several

(2) Number of locations: - one - several

(3) Input character: - deterministic - stochastic

(4) Output character: - deterministic - stochastic

(5) Dynamics: - static - dynamic

(6) Objective: - optimizing - reliability equation - descriptive

(7) Mechanism (ordering rule) : The mechanism (ordering rule) can be defined by giving answers to the basic questions of inventory control as follows: When to order? - at every t time units - when the inventory is equal to or below

s quantity units How much to order? - 4 quantity units - an amount which brings the inventory

to level S The inventory control mechanism is de- fined by the possible combinations of these parameters, i.e. there are (t,q), (t,S), (s,q) and (s,S) ordering rules (see Naddor [ I]). When one of these param- eters is prescribed (i.e. it is not to be determined by the model) then an index p is used with this parameter. Considering the relative frequency of the various

mlsrirtg rules, we have detined the fol- bWiJ1~ code vah~es:

4r.y 1, u,,y 1. U,clp) cG$d

. !!p,S)

- csl

- ~sg,q) - hf) or (s,qp) - Optic - W) or (s&)

48) 8Wiewing period: no review discrete time review continuous time review

(9) Handling of shortage: - rhortage not allowed .-- backtogged orders _ lojit orders

ClO) Lead time: - no lead time

caMant lead time - deterministic, variable lead time

raEdom lead time - - partial or continuous deliveries

We u* three general code values which can appear in connection with any of the 45 codes:

-1:

7: 9:

the question of the code does not make sense in the given model the question is not treated in the model other -~ this is always followed by a remark.

Obviously, for a complete description uf the contents of these r&es more explanation would be necessary (we have specific inter- pretations of all codes) but we only wanted to @ve an impression of what this code system b~ks like. Nevertheless, we must state that there are many cases when some given features of a model cannot be descried Car cannot be fully described) with these

es, so we had to use the code value 9 ‘? quite often, and/or debated the

, the identification number of the bibliographical spedfica-

tion form a vector of 57 p:acitiuns which Jc- scribes the model. These data have been computerized.

Apart from creating this record of all models, we have also produced a description of the models (between l-3 typewritten pages). Ail the descriptions have the very same structure, namely, besides the identifica- tion part they consist of the following three parts:

-_ conditions considered in the model _- objective function - algorithm of solution The objective of this description is to make

information about our model library available also to those who do not have access to our computerized system, and also to those who are not interested in those details (e.g. managers as potential users).

With the 57position record of the 336 models in the computer, we first needed a program system which makes the model system available for the users. The program system has two basic functions:

-- It must be able to print models meeting any giver, specifications (what i; a sort of “librarian” function) which is necessary both for scientific and practical applica- tions.

- It must be suitable for the statistical evaluation of the model system.

3.2 Statistical evaluation of our sample of models

Having created this program, we started the evaluation procedure. First, we carried out the classical statistical examinations, namely we printed the frequency matrix, relative frequency matrix, contingency tables, conditional frequency matrices and condi- tional contingency tables.

It was a very important (and multi-step) problem to decide what variables (codes) are to, be used for the contingency tables and what conditions are to be defined for calcu-

lating conditional frequencies and contin- gencies. The selection was first based on our a priori judgement of the importance of the various codes and then on the findings of the first evaluations.

Parallel with this work we have made preparations for using multi-variable statistical methods (i.e. factor and cluster analysis), and carried out many computer runs. In our analysis, the models characterized by code values are regarded as observations described by characteristics measured on a lower scale. Our models are points of a state space deter- mined by codes. Having analysed the struc- ture, shape and properties of the point clouds which consist of our models we can map our system and learn its regularities. The impor- tance of these is that they treat the char- acteristics of the models simultaneously.

We have alrealdy completed the factor analysis. It has shown that our code system consists of’ fairly independent variables and that the structure of models is influenced at most by the following factors (codes): ordering rule: deterministic vs. stochastic character of input and output; time phasing of input and output; quantitative character- istics of demand (discrete vs. continuous, one or several lots, etc.); and the cost factors considered.

We also made over a hundred cluster analysis computations with different methods - various procedures of the convergent K-means, Forgy and McQueen methods - and different subsets of the variables. The results of the cluster analysis provided a good basis for the classification of the models.

We are well aware of the limitations of using these methods for a problem like ours but, handling the results with special care and caution, we can avoid the danger of mis- leading conclusions.

That is the current state of our research, which is, of course, only a station in the pro- cess of our work, which is a multi-step proce- dure with many feedbacks, including the

involvement and analysis of new models, the continuous improvement of the capabil- ities of the computer program, the extension of the analysis of the existing models and, of course, the investigation of the applicability of our system in practice, which will be dis- cussed later.

4. SOME CHARACTERISTICS OF THE SAMPLE OF MODELS

llaving made the elementary statistical analysis of our sample, we could derive some conclusions about the connections among the various properties of the models. Obviously, the properties of models form the basis of their groupings, since we put in a specific group those models which have similar char- acteristics from some points of view. We have found that it is much more purposeful and important to establish groups (subsets) of models which might have common ele- ments (i.e. a model can be a member of several groups), than to try to find separate subsets, which is the common approach in the literature. We felt that our grouping is a more appropriate one from both the point of view of theoretical analysis (since the common appearance of the various conditions, features of the models can be better observed and examined) and practical application (it facili- tates the search for a model with particular characteristics).

4.1 Results of the analysis of frequency tables

As a result of the analysis of the frequency matrix, we could discover the primary and basic features of the model sample. (Here we can give only results regarding the distribution of values of the main codes.)

We have found that the great majority of the models handle a single item and single location. Models of multi-location systems are especially rare.

It is interesting that the handling of input

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and output processes is closely connected with the economic system of the respective author’s nation. Namely, for market econ- omies inventory models with deterministic input and stochastic output are charactetistic, while for socialist economies those with stochastic input and deterministic output. Thjs phenomenon reflects the differences in the economic conditions of market and planned economies. As the majority of models are from western countries, for the whole sample of models the first input- output version is typical.

Static models turned out to be much more common than dynarqic ones. Hcwever, it

t be kept in mind that there is no general- cGepted defmition for “dynamics”, so tication of a particular model is not un-

zquk-ocal. Nevertheless, we are probably right in stating that authors prefer simpler treatment of connections in time.

In anatysing the objective of models, we have distinguished those with objective functions, reliability-type models and descrip tive ones. Not surprisingly, most models are cost-optimizing. We still have a number of models with reliability constraint. (These models chiefly originated from the Hungarian inventory literature.) And there are a few descriptive models.

As far as zhe ordering rule is concerned, modefs of the (s,q) type proved to be most frequent. We have collected 104 models of that type.) There are a great number of mod&s of the (t,S’) type (91). The number of WI models is 60, and that of (t,q) type is 54. We also have some special models, which cannot be put into any of these “‘classical” groups.

The ratio of models with periodic review and those with continuous review is 2 : 1, This

dots that the assumption of a non-constant review is certainly more realistic. In the

part of the sample, occurrence of is allowed, and demand is back-

fbwe~er, there are a considerable

number of models with no shortages and with lost sales.

In more than 50% of the models there is no lead time. If lead time is considered, it is constant in most cases, and there are only a few models with stochastic lead time.

4.2 Results of analysis of contingency tables and relative fuequency matrices

While in the previous part we have char- acterized the sample of xnodels by the distri- bution of individual codes, we now turn to the examination of connections among the various characteristics of the models. Some aspects of the analysis are as follows.

Complexity

We have analysed the effects of three properties which obviously make the models more complex. These properties are multi- item, multi-location and dynamic character, as opposed to single-item, single-location and static character. These properties appear in the main codes 1, 2 and 5, as we have de- scribed earlier. It is quite insignificant that the price of the introduction of any of these more complex conditions (which are, in prin- ciple, to reflect more complex processes in practice) is that in many other respects the model will be simpler. If any of the three properties are treated in the model, generally one will find that

- there is no lead time in these models - input is not treated at all, or it is deter-

ministic with deliveries in one lot - there is a prescribed order period.

Besides these, the following further properties in multi-item and/or multi-location models can be found (these two groups of models have pretty much the same structure):

- they are mostly deterministic in their input and output

- they have simple ordering rules - in many cases shortage is not allowed

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(this is in connection with the first property here).

Most of the multi-item models are static, while there are a lot of dynamic models among the multi-location types. Dynamics attracts other different properties. For our models generally

- input is deterministic, output is stochastic

- there is a discrete reviewing period - the mechanism (ordering rule) is of the

(0) type - shortage is almost always explicitly

treated - they are cost-optimizing models.

It seems that dynamics not only allows but calls for a more complex system of conditions - probably since under simpler conditions it is just not worthwhile or not interesting to introduce dynamics.

Objective function and mechanism

Here we have collected the findings con- cerning the main codes 6 and 7. We have found that there is a group of mechanisms, n.amely (t,q), ($,q), (s,,q), which is char- acteristic of the elementary models. In these models generally:

- input and outtut are deterministic - shortage is not allowed - reorder (scheduling) periods are equal.

These simple mechanisms can be also found in some of the models of very high complex- ity from other points of view.

Models of the 0,s) type have, normally, continuous demand and orders usually filled in one lot.

More complex models are built usually on the (s,q) and (s,S) mechanisms. These models have generally :

- stodhastic output - continuous review of inventory level - backlogged orders (i.e. in case of short-

age the demand is recorded and filled when the next shipment arrives)

- nay-prescribed and non-equal reorder periods.

For (s,q) models it is characteristic to have a positive lead time and demand occurs at random intervals, while in (s,S) models ship ments normally arrive in one lot.

The vast majority of models are cost-mini- mizing, so the mechanisms usually accompany that type of objective function. There is one exception. We have found a special group of models with the objective of slitirfaction of a reliability equation (expressirig the require- ment of obtaining a given service level) instead of minimizing a cost function. This set of models is, with very few exceptions, of the (t,S) type, and has the following character- istics:

- stochastic input - static feature - there is a reliability constraint for

handling shortage - both input and demand are either con-

tinuous or occur in several lots. There is also a group of descriptive mod.els in our sample. They are mostly determir;sstic, but characterized by some more complex fea- tures, e.g. multi-item, multi-location and/or (s,q), (s,S) mechanisms.

Input and output processes

One of the features having the most impor- tant influence on the structure of models is the character of output. Models with deter- ministic output have - again, in general - characteristics very different from the ones with stochastic output. The characteristics of the most common models can be seen in Table 1.

As for the input, most of the models are deterministic, Those models with stochastic input are mostly of a reliability type, the characteristics of which are described in the previous section. The cost-minimizing models with stochastic input are generally based on the (s,q) ordering rule.

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TABLE 1

~~n~~~~n between the character of output and other critics of inventory models

putis deterministic stochastic

~_~~-Lcc~-~-- _.--

no

not allowed

zero (Of input arrives in several I.4514 or it is con- timmus)

vhy different

types

esntitttxnls

WI, b.9). 0,s)

at d’rccrrte interMs

allowed

non-zero

Another characterization of models, con- nected with input, can be based on the fact that continuous review and probabilistic lead time normally go together. Sotie of the fea- tures of these models are as follows:

-

--

-

-

_.

in many cases both input and output are stochastic most of them are of tihe (s,q) type order period is not prescribed and not equal orders are backtogged in case of shortage decision variables are discrete input is random in time demand occurs (often unit demand) at several random points in time unit costs of the model have some particular characteristics

in one lot

in one Iot

holding, ordering. shortage

._____- .-_I- H

__ there are several special external con- straints,

Inventory models

r

titnr of inventory models,

13.5

4.3 Classification of the models

We consider the classification given in Pig. 1 as the main theoretical result of our research. There are a great mmber of classifications of inventory models in the literature (see, for example, Hadley and Whitin [ 21, Naddor [ 1 I, Veinott [ 31, Hochstgdter [ 4 j, Kiemm-Mikut [5], Aggarwal [6], Tersine [7], Hollier and Vrat [ 81). While these classifications are suitable from many points of view, we blelieve that for a really well-founded general clas- sification there is still more research needed. We do not think our classification superior, as it also has several limitations; for example, it is based on existing, not theoretically pos- sible, models, but it is a good basis for further theoretical analysis, and for the creation of an implementable computerized model library.

The classification is given in Fig. 1, which gives the 12 main classes of models and their subgroups. We must add that the formation of these subgroups is not as defensible as of the main classes, since in a subgroup there are normally only a few models. As can be seen, the aspects of forming the subgroups are not unified (in contrast to the main classes), but depend on the special characteristics of the classes.

We believe that we still have a great deal of work ahead of us before we can finish our research. Nevertheless, we hope that the above allows the reader to gain an insight into our research.

So far, we have reported on our theoretical results, now we turn to the field of applica- tion,

5. COMPUTERIZED LIBRARY OF INVEN- TORY MODELS AND ITS USAGE

When we have defined the objectives of our research, as we have emphasized at the begin- ning of this paper, the potential practical application was at least as strong an incentive as the expected theoretical findings. The first

and basic result of our work from the point of view of applicability is the knowledge of the 336 models. We have created a model library where these models are available to those who wish to use the models. Our model library consists of four subsystems:

1. The library of textual descriptions of models

From the textual description, one can see the main characteristics of a model, even if one is not a specialist of models. There is no need for any special explanation to use it; this library of description can be given to any- orie interested. There is, of course, an exact reference to the origin of the descrption, i.e. to the complete, original model.

2. The library of code description of models

This library has two forms: - the system of code sheets filled out

manually. There are both code values and remarks and references on these sheets:

-. the computerized code system, which kas been created by an appropriate trans- formation from the above form, ex- cluding, or transforming, of course, most of the textual information. This is the heart of our library; it can be con- sidered as the most important subsys- tem. As we have indicated earlier, we have created a computer program which is able to answer many different questions and thmugh which we have access to models of various types defined from various points of view.

3. The library of flow charts of models

This is to create a connection between sub-, systems 1 and 4. This is for technical purposes, to indicate the usage of the various models.

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4. TheI oftJpmahg~8

consists of computer programs of thm to determine optimal values of

decision variables of the various models, that is, directly for practical usage.

For our 336 models we have completed libraries 1 and 2, and we have partly estab-

We have in mind the following method of plication. Let us suppose that a manager

ecides that he wants to use our model library fy optimal parameters for a given

t. We is to turn to the “‘librarian” who a questionnaire, which corresponds

to our code system, but it is written in a for communication be- tice and theory. On the

of this questionnaire, the model(s) g properties specified by the manager is

(are) requested from the computer. If there are exactly such models, the search procedure is fiiished. Otherwise, the computer will print the models “closest” to the requested one (i.e. models differing from the requested one in one, two, etc., codes). Then there msrst be a discussion on the applicability of the models selected. If all involve;5 can agree, the right model can be called from library 4. It can be modified if necessary, with appropriate input parameters put into it, and thus opGmal values of the decision variables can be obtained. (We do not deal here with questions of implementation - for example, availability of data, putting optimal decision

es into force, etc. - which are of basic mrportance, but are not closely connected

problems of our research.) e hsve already carried out an application

t of our existing models, which built into the inventory control of a complex computerized

created by the ciation of the

Chemical Industry, which is a joint ve:nture of 14 Hungarian companies in the chemical industry. So far our results are quite promising.

REFERENCES

As for the resources of models, only books are given here - the complete list of references contahs 146 further items, articles of journals, research reports, etc.

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2

3

4

5

6

7

8

9

10

11

12

13

14

15

Naddor, E. (1966). Inventory Systems, John Wiley and Sons, Inc., New York. Hadley, G. and Whitin, T.M. (1963). Analysis of Inven- tory Systems, Prentice-Hall. Velnott, A.F. (1966). The Status of Mathematical Inventory Theory, Management Science, 12(11). Hochstiidter, D. (1972). Stochastische Lagerhaltungs- modelle, Springer-Verlag, Berlin. Klemm, H. and Mikut, M. (1!372?. Lagerhaltungs- modelle, Verlag die Wlrtschaft, Berlin. Aggarwal, S.C. (1972). A Review of Current Inventory Theory and its Application, International Journal of Production Engineeriug, 443-482. Terslne, RJ. (1976). Materials Management and Inven- tory Systems, North Holland, Amsterdam and New York. HoBier, R. and Vrat, P. (1978). A Review of Multi- Echelon Inventory Control Research snd Application, University of Bhmingham. Chlk&, A. (ea.) (1%). The Economics and Manage. ment of Inventories, Proceedings ,Jf the First Inter- natioti Symposium on lnventorles. Elsevier, Amster- dam and Akademlai KladB; Budapest. Arrow, KJ., Karl@ S. and Scarf, H. (1958). Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford. Buchan, J. and Koenigsberg, E. (1963). Scientific Inven- tory Management, Prentice-Hall, Englewood Cliffs, N.J., Naucsnoje upravlenyije zapaszami, Nauka, Moszkva (1967). Pr&opa, A. (ed.) (1973). Colloquia Math. Sot. J. Bolyal, 7, Xnventory Control and Water Storage, Gyiir, 197 1, North Holland. Rizslkov, Ju.L (1969). Upravlenyije zapaszami, Nauka, MOSZkW.

scarf, H-E., Gilford. D.M. and Shelly, M.W. (1963). Multistage Inventory Models and Techniques, Stanford University Press, Stanford. Wagner, H.M. (1968). principles of Operations Research, Prentice-HaR, Englewood Cliffs, N.J.