The Use of Flow Model Analysis (FMA) in the Case of Incomplete Mathematical Models

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

THE USE O F FLOW MODEL ANALYSIS ( F'MA ) IN THE CASE

OF INCOMPIEIX MATHEMATICAL _MODELS

a e g &rdakov, h n a t o l i G o l o v i n , K l i m Kim, A l e z a n d e r h n o v , Igor % m p o l ~ v *

A l e z a n d e r h n o v ^, E d i t o r

September 1984 CP-84-40

Dr. Anatoli Golovin and Dr. Alexander Umnov, IIASA Dr. Oleg Burdakov, Computing Center of t h e Academy of Sciences of t h e

USSR,

Moscow.

Dr. Klim Kim. central Economic-Mathematic?! institute, Moscow.

Dr. Igor Shompolov, Moscow Physico-Technical Inst;.l,te, Moscow.

C o l l a b o r a t i v e P a p e r s

report work which has not been performed solely at the International Institute ,for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organi- zations supporting the work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS

A-2361 Laxenburg, Austria

This paper describes part of an investigation made by I M A in cooperation with

UNIDO

and a number of institutions of the Academy of Sciences of the

USSR.

The main purpose of this collaboration is to develop new methodologies for analyzing mathematical models and to test them on applied problems of practical value.

Vi tali Kaftan ov

Deputy Director of IlASA

Summary

The main aim of the 'System for Analyzing Mathematical Flow Models' ( F'KA system ) described in this paper is to supply the decision-maker with a computerized tool for quantitative investigation of problem t h a t can be described, a t least partially, in terms of standard mathematical models of the Bow type.

The F'MA system permits the decision-maker to find states of t h e con- sidered model t h a t satisfy all introduced constraints, are close t o t h e desirable structure, and are optimal with respect to single- or multiobjective evaluation.

1. Introduction

The most important problem in mathematical modelling consists in evaluating t h e reliability of t h e results obtained with t h e models. Usually t h e reliability of t h e results is assumed t o be g u a r a n t e e d by t h e adequacy of t h e model, i.e. by t a h n g into a c c o u n t all essential relations defining t h e behavior of t h e object being modelled. Therefore, i t is reasonable t o a t t e m p t t o solve simu- lation, optimization, or forecast problems where t h e models are considered ade- quate.

I t would be naive t o believe, however, t h a t t h e m a t h e m a t i c s t h a t h a s been developed intensively during t h e last hundred years enables us t o descr'_be adequately all properties of objects (e.g. socio-economic phenomena) whose essential n a t u r e differs from t h a t of t h e objects of physics o r engineering.

Practice indeed confirms t h a t our mathematical culture is well developed for describing physical p h e n o m e n a a n d engineering operations.

Disregarding t h e problem of constructing adequate m a t h e m a t i c a l models, we shell consider h e r e t h e possibility of using incomplete models, which take into account only some of t h e essential relations a n d properties describing t h e object; t h a t is, in cases where construction of an adequate model i s impossible or extremely difficult. The main purpose of this paper i s t o show for a sufficiently wide class of decision-making problems t h a t t h e completeness of a model is n o t necessary for r e s u l t s obtained with the model t o be correct.

Such a model may, for exampie, be used n o t for seeldng t h e "best" decision but r a t h e r for establishing whether some decision is acceptable or not from t h e viewpoint of t h e decision-maker. Indeed, if an incomplete model identifies some decision a s unacceptable ( i.e., t h e decision does n o t satisfy some of t h e formal conditions of acceptability included in t h e model ), t h e n this decision will be also unacceptable for any more complete model. At t h e s a m e t i m e it is

possible t h a t an acceptable decision obtained from an incomplete model may be unacceptable for a more complete model. When we say that one decision is more acceptable than another, i t means t h a t the first decision satisfies all of the acceptability conditions m e t by the second as well as some additional acceptability conditions.

Thus, in t h e approach t h a t we propose for utilization of incomplete models, the mathematical model, in conjunction with the computer, is used in decision-making only as a tool for picking out a set of acceptable solutions. The decision-maker, using informal, empirical, or intuitive criteria, chooses the decision t h a t is t h e best from his point of view.

As well as leading to satisfactory results, an advantage of this approach over traditional ones consists of the possibility for the decision-maker to have a more active role. On the other hand, as will be seen below, in t h e proposed approach it is necessary to avoid certain difficulties arising in its practical realization.

2. A Typical Problem

This section is devoted to t h e analysis of trade markets, for which ade- quacy of a mathematical model may be proved easily, and which demonstrates the great potential of the described approach.

A system of partners ( e.g. private persons, companies, countries, regions and so on ) tradmg in a s e t of commodities within a given period is called a ha& m a r k e t . If the v o l u m e s and p r i c e s of the commodities a r e known, it is possible to evaluate e z p o r t , i n z p o r t , and b a l a n c e data

characterizing t h e state of the trade market.

Using these basic data, one may evaluate the level of acceptability of the current state of the trade market from the viewpoint both of each of the partners and of the market as a whole.

The definition of desirable or acceptable states of the market permits u s to formulate the following questions :

-

Is t h e current state of the market a desirable one ?

-

If not, how far is t h e current state from t h e desirable one ?

-What should we do to bring these two states nearer t o one another ?

Let us s t a r t by describing a mathematical model of the trade market.

Let vt. be the volume ( measured in physical units ) of t h e k t h commodity sold by the i t h partner to the j t h one. If the unit price of this commodity is

p k

, we may defme the export, import, and balance for t h e total trade between the partners as

impU

=

e z p i i

where

K

is the total number of commodities traded.

The total volumes of exports and imports for the i t h partner will be

IMP,

= f

^WnpG

j = 1

where N is the number of partners, and fmally

IMBALANCE; =mi

-IMPi

It is very easy t o prove t h a t t h e sum of al1,exports equals t h e s u m of all imports using to t h e above relations.

Let us suppose t h a t one may define lower and upper acceptable bounds for the export, import and balance indicators for each of t h e partners.

We wiU call a s t a t e of the trade market acceptable if the constraints

EXP.

^c

E P , E P ,

-

-'

^-

IMP. ^c IMPi C I ~ P ~

'

^-

IMBALANCE, ^g IMBALANCE, IMBALANCE,

are valid, JOT all i=[l,N]

The values of t h e lower and upper bounds may be decided by experts accordmg to the scenario t h a t is going to be considered. For example, data for t h e i t h group of constraints may be defined by authorized representatives of the i t h partner.

Besides t h e data characterizing t h e overall t r a d e balance of each partner, t h e r e may also exist constraints due to limited industrial capacity, transport capabilities, and so on. Therefore, t h e system of constraints describing the acceptable s t a t e s may often b e augmented by supplementary inequalities, for example, of t h e following type:

v t _I ^g ,,.t. _Y ^g

ck.

_v

f o r a l l k , i and j

.

I t should be emphasized h e r e t h a t the ezpert opinions expressed i n t h e constraints described above may sometimes appear t o be far from realistic, or even inconsistent. Therefore we m u s t be ready t o tackle cases where t h e r e is no acceptable s t a t e a t all. On t h e other hand, i t is also possible t h a t t h e r e will be many acceptable s t a t e s of t h e trade m a r k e t for t h e same s e t of constraints.

We can now use our definition of an acceptable s t a t e of t h e t r a d e m a r k e t t o evaluate how far a given s t a t e is from a n acceptable one.

Let us assume for the moment t h a t all conditions a r e consistent. Then there exists an acceptable s t a t e for the model under consideration. We can transform a given s t a t e into this acceptable one by making appropriate changes in t h e volumes of commodities sold.

If

this transformation involves adding zt t o the volumes v$

.

t h e n the relative value of this change,

characterizes t h e degree of imbalance for t h e flonr of t h e k t h commodity from partner i to partner j . The absolute value m u s t be used here because z$ may be either positive o r negative.

One measure of t h e "unacceptability" or "imbalance" of t h e s t a t e of the market as a whole could be formulated as

This evaluation of t h e "distance" from t h e given s t a t e t o t h e acceptable one only has practical value if the acceptable s t a t e is unique. But usually the acceptable states a r e in fact nonunique and a different p value is associated with each.

One way a round this problem is to take just t h e minimum of these p values, t h u s eliminating t h e ambiguity in our definition of "acceptable". In

other words, we define t h e difference between the given and acceptable states as

, min

P

-

2 ~ ( 2 )

-

^{min max}

p

-

² [ k , i , j & S

[+]I

m e v a l u e of p * s h o w s w h a t minimum r e l a t i v e c h a n g e .is r e q u i r e d t o t ~ ~ ~ r n f o r m t h e g i v e n s t a t e of t h e t r a d e m a r k e t i n t o an a c c e p t a b l e o n e .

Mathematically, t h e procedure for finding t h e imbalance p , which is in fact a special case of t h e 'Chebyshev approximation problem, can be reduced to t h e

following m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m . Minimize p

with respect t o

1

^p,

zt,

^{for all} i , j , k j subject t o

for all i , j , k

EN', 5 E m i

s

Expi -

- -

IMP. < IMPi S IMPi

-'

^-

IMBALANCE,

<

IMBALANCE, ^S IMBALANCK where

IMBALANCE, =EXP, -IMPi,

f o r all i = [ l , ~ ]

.

3.

The General Flow Model

We shell now consider a n o t completely connected graph of n o d e s linked by oriented flows. All nodes are n u m b e r e d and each of t h e nodes can be a source, a drain, or simultaneously both

.

Each of t h e flow may consist of different t w e s of flow o r components. Figure 1 shows an example of this graph.

Let t h e model have N nodes and K types of fiow. The main quantitative characteristic of a flow is i t s value. Generally, each type of flow h a s i t s own unit measure of value. There may also exist a common m e a s u r e of t h e values of all now types, which is called t h e equivalent value or simply equivalent.

If

t h e donr from t h e i t h node t o t h e j t h node of t h e k t h type is u t , then its equivalent will be

where

p$

is a given positive constant.

The bow model analysis s y s t e m uses t h e values us and e$ t o descri! e both input and output d a t a of t h e Aow model.

Each s t a t e of t h e model, i.e. nonnegative cube m a t r i x with elements u s , can be specified as unacceptable or acceptable or desirable. The unacceptable s e t consists of those s t a t e s of t h e model for which a t least one of t h e necessary conditions of acceptability is violated. The complement of this s e t i s the s e t of acceptable s t a t e s . The s e t of desirable s t a t e s of the model is described by condi- tions t h a t are not necessary. Therefore, this s e t can have intersections with both t h e acceptable and unacceptable sets.

To simplify the description of t h e definition of t h e acceptable or desirable s t a t e s of t h e model in t h e FbIA system, we can use t h e following a d i t m y vari- ables

N N K

C C C

ei:...

Sometimes we may have the same values of

pk

for all feasible indices i , j , k . In these cases t h e following variables can be also used in t h e statement of the problem :

Summation for a subset of feasible indices is n o t permitted in an explicit way here, but it is always possible to split any node into a system of new ones and extract desirable subsets of indices.

Finally, the user can formulate the problem in terms of t h e imbalancing variables. They a r e :

Imbalancing variables m a y have both positive and negative values.

4. The Conditions of Acceptability

The user of t h e FMA system may define t h e conditions of acceptability of a s t a t e of the model by introducing constraints on t h e variables

v h ,

e$ and all auxiliary variables.

The constraints on t h e absolute value of a variable may be of t h e follon7ing types:

-

t h e variable m u s t be equal t o a given value,

-

the variable m u s t be not less t h a n a given value,

-

t h e variable m u s t be not g r e a t e r t h a n a given value,

and ( subject t o t h e initial value is given )

-

t h e variable cannot be changed,

-

t h e variable m u s t not decrease in value.

-

t h e variable m u s t not increase in value.

It is also possible t o introduce constraints for t h e r a t i o of a pair of vari- ables :

-

t h e ratio m u s t be equal t o a given value,

-

t h e ratio m u s t be not less than a given value,

-

the ratio m u s t be not greater t h a n a given value.

In t e r m s of t h e equations and inequalities these c o n s t r a i n t s may be writ- t e n for a variable v a s

where C is a given constant.

Analogously, for a pair of variables v 1 and v e

Finally, in t h e FMA system we can use the most general allhe dependence between two variables

where A and

R

are arbitrary constants.

All variables described in Section 3 can be included in these relations. The total number of constraints is limited only by t h e available computer resources.

5. The Conditions of Desirability

The simplest way to introduce a desirable state into this model is to describe it explicitly. The user may define the desired value for any subset of flows in the model. The F'MA system proves whether this definition is acceptable one or not. If the definition is acceptable, the system will calculate appropriate values of t h e remaining Bows to grant t h e acceptability of the state as a whole.

Normally the desired state is unacceptable, i.e. the constraints describing the conditions of acceptability are incompatible with the desired values of the flows. In this case the

FMA

system builds a new state of the model that is acceptable and is the closest to the desired state in the sense of the Chebyshev metric.

Let a considered acceptable state be v and the given desired s t a t e be v * .

Then we may define t h e distance between these two s t a t e s as ,,

,*

-

^,,k.

-

min m a x abs v

'J

p - v a , j , k vij ^{k *} ' subject t o v satisfying all of the above-defined constraints.

This minimax objective permits us t o avoid ambiguity in t h e solution.

In t h e FhdA system a special modification is used. Very often t h e chosen m e t r i c depends only on a subset of AOH'S, which we call lending flows. The remaining 8 0 ~ ~ s may have arbitrary values, which have no practical meaning.

I t

is reasonable t o try t o continue t h e minimax procedure, fixing all t h e leading flows a t t h e optimal levels.

In practice, this m e a n s t h a t all leading flows a r e removed from considera- tion as variables in t h e optimization problem and a new s e t of leading e l e m e n t s ( with a new value of p ) is built. This s t e p may be repeated until all flows a r e fixed or p becomes zero.

This procedure of s e q u e n t i a l f i z a t i o n produces a ranking of t h e whole s e t of dows in t h e model. Let pf be t h e solution of t h e optimization problem in t h e t t h s t e p of t h e fixation process and

Qt

be t h e correspondmg subset of leading flows. Then pf may be t r e a t e d as a relative measure of t h e required r e l a t i v e change of t h e flows in t h e subset

Rt

t o bring t h e m t o t h e given desirable s t a t e .

In t h e FMA system t h e u s e r can control t h e sequential k a t i o n procedure, limiting t h e number of steps i n t h e fixation or terminating i t a s soon a s t h e required level of p is reached. The fixations a r e of course made simultaneously for all defined constraints.

Finally, t h e FMA system permits us t o use weight c o e f f i c i e n t s t o c o r r e c t t h e dependence of p on t h e flows i f necessary. In t h e general case t h e distance between t h e acceptable a n d desirable s t a t e s is

v&*

- ,,,

k m i n max abs v

P

= % .

i , j , k vk* '

v

where w$ is a nonnegative c o n s t a n t .

6. Optimization in the FMA System

The FMA system with t h e features described p e r m i t s us t o solve optimiza- tion problems, maximizing or minimizing any of t h e variables introduced in Section 3. To do this i t is sufficient t o include in t h e considered model a new formal flow t h a t equals t h e optimized function. We shall call this t h e o b j e c t i v e bow.

By giving t h e desirable objective fiow a very large value ( for maximization ) or a very small value ( for minimization ) a n d choosing an appropriate weight coefficient for t h e flow, we shall find t h a t t h e resulting value of t h e objective bow is optimal.

In t h e same m a n n e r we can carry out a multiobjective optimization pro- cedure, by introducing several objective flows and supplying t h e m with equal weight coefficients. A point of t h e Pareto s e t will be t h e solution in this case.

7 . Andping Structural Change with

the

FXA System

Any feasible combination of t h e above procedures, which manipulate t h e weight coefficients and objective variables, may be used in t h e FMA system. One of t h e most important procedures in practice is t o i n s e r t null Bows. Of course, direct use of a flow with zero value i s not possible, b u t t h e r e can be no objection t o do this if t h e flon7 h a s zero weight. This makes i t possible t o reserve a new element in t h e considered model.

The reservation of null flows may be useful in improving t h e model in t h e case of infeasibility. An unacceptable, but desirable, s t a t e of t h e model may be approximated by an acceptable s t a t e t h a t is found by t h e FMA system minimiz- ing t h e 'distance' between t h e s e states.

Finally, t h e F M A system can be used to analyze dynamic models. In this case t h e desirable s t a t e of t h e model is considered as t h e initial state. The con- ditions of acceptability describe the final state. If necessary, t h e dynamic pro- cedure m a y consist of several steps, each of which h a s an independent descrip- tion of t h e conditions of acceptability. The final s t a t e for one s t e p is used a s t h e initial s t a t e for t h e n e x t step.

0. An Example of Using the

FMA

System for Energy Development Projections We shall now t o demonstrate how t h e F'MA system can be used t o analyze t h e development of t h e CMEA energy market until t h e year 2000. Usually, such m analysis involves detailed considerations of the fuel-energy balances and t h e energy-economy interactions in each of t h e CMEA countries. These subjects are described by m e a n s of models t h a t take into account a large n u m b e r of pararn- eters. Generally, t h e r e is a g r e a t deal of uncertainty attached t o these pararne- t e r s with r e s p e c t t o f u t u r e developments. One possible way of describing a l l t h e essential features of t h e modelled system is t o use analytical techniques t o define more or less realistic trajectories of f u t u r e energy developments.

Nevertheless, t h e r e will still be problems of model verification, data reliability, and t h e like. Besides, t h e m o r e parameters t h a t are used in t h e model, t h e more difficult i t becomes t o r u n t h e model, t o analyze t h e results, and t o elim- i n a t e t h e e r r o r s .

Another approach h a s been found suitable for assessing future develop- m e n t s of energy systems, nrhich takes into account t h e acceptability of t h e f u t u r e situation of t h e modelled system. This approach will now be described.

The

FMA

s y s t e m was used t o assess some of t h e boundaries of an acceptable s t r u c t u r e of primary energy consumption for t h e European countries of the

CMEA,

based on estimates of likely trends in t h e production of primary energy sources a n d on assumptions regarding future r a t e s of economic growth and

energy elasticities for these countries over t h e period 1985-2000. Feasible values ( or ranges of values ) were then identified for energy exports from the CMEA countries ( mainly from the Soviet Union ) t o the r e s t of t h e world.

The following assumptions were made:

-

F u t u r e energy imports to t h e CMEA will not exceed existing ones a n d have to be minimized.

-

Assuming t h a t a s e t of feasible solutions exists, t h e process of finding an acceptable solution has t o take i n t o account

t h e criterion of minimizing a maximal change in t h e strii ;ture of energy consumption; t h a t is, of finding a feasible s t r u c t u r e for t h e f u t u r e t h a t is as close as possible to the existing s t r u c t u r e .

-

Energy flows between

CMEA

countries have t o be as stable as possible but, a t t h e same time, export of energy from these countries t o t h e r e s t of t h e world h a s t o be maximized.

The process of assessing the acceptability of t h e future s t r u c t u r e of pri- mary energy consumption has t o include two m a i n procedures:

-

assessing t h e existence of a feasible s t r u c t u r e ,

-

defining possible boundaries of acceptability.

These procedures were performed for each five-year interval of t h e period considered, and t h e results of one step were used as t h e initial conditions for t h e next step. The first procedure consisted of finding a feasible s t r u c t u r e of energy flows subject t o t h e criteria mentioned above; t h e second procedure involved t h e investigation of variables.with values close t o t h e previously found

solutions for the s t r u c t u r e of primary energy consumption for each country.

One of the important results of the second procedure was a definition of possi- ble ranges of energy exports from the CMEA countries t o the rest of the world.

A substantial feature of t h e acceptable s t r u c t u r e s of primary energy con- sumption in the CMEA is t h e changes in the shares of solid fuels. For most coun- tries, these shares have to be decreased. But in the case of Hungary, i t is possi- ble to have the same share of solid fuels in 2000 as in 1980; and in t h e case of Roumania this share has t o be increased.

The share of liquid fuels has to be decreased almost everywhere (except in Poland). For gaseous fuels, t h e share has to be incrrased in every country ( except Hungary and Roumania ). The substantial growth of the s h a r e of pri- mary electricity in total energy consumption is caused by the development of nuclear energy programs in t h e

USSR

and the East European countries.

Under t h e s e changes in t h e s t r u c t u r e of primary energy consumption, t h e possible a m o u n t of energy exported from the

CMEA

countries was assessed. It was found t h a t t h e main energy source t h a t could be exported is natural gas.

The results of the study a r e only preliminary and a r e based on data and assumptions about future energy development t h a t had been made by the authors themselves. Therefore t h e results serve only t o show t h e feasibility of the approach proposed for t h e assessment of acceptable s t r u c t u r e s of primary energy consumption and exchange within t h e CMEA region and also for estima- tion of possible primary energy export levels by t h e CMEA including exports of natural gas.

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Sequential U n c o n s t r r n e d Minimization Techniques. John Wiley, New York.

Issaev, B. and Umnov,

A.

(1982). Integrated Economic Balance of a

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Lenko, M. (1983). System of

FORTRAN

Subroutines to Analyze

Trade Markets (TMA) : User's Manual and Programmer's Guide.

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