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Modelling decentralized decision making in the energy sector
O.tlEG.4 [nt J. or" Mgmt Sci.. Vol. 12. No. 5, pp. 437~.4 "~. 1984
0305-0483 8.-t8 3 . 0 0 - 0 . 0 0 Copyright ~ 1984 Pergamon Press Ltd
Printed in Great Britain. All rights reser~,ed
Modelling Decentralized Decision Making in the Energy Sector J - E M M A N U E L SAMOUILIDIS A ARABATZI-LADIA Energy Policy Unit, National Technical University of Athens, Greece (Received November 1983: in revised form January 1984) Large linear programming models, which have been widely used to determine the optimal structure of national energy systems, are based on the assumption that there is an absolutely centralized decision making process within the energy system. In this paper an attempt is made to match the real decision structure of a given energy system, by decomposing an LP energy model into smaller models, with the corresponding system decision centres. This is done by applying the 'transfer price' algorithm of Dantzing and Wolfe. The 'master' problem corresponds to the central planning unit, i.e. a Ministry of Energy, whereas the subproblems correspond to peripheral operating units, i.e. enterprises, usually state owned, which produce and distribute the energy carriers. The optimal plans of the peripheral units are submitted to the central unit, which through the mechanism of pricing of both common resources, inputs and energy services outputs, co-ordinates the overall planning of the energy system. An illustrative example is given referring to the Hellenic national energy system. The research reported is placed within a wider research endeavour, whose objectives and main line of work are also given.
INTRODUCTION PROBLEM O f suboptimization, e n c o u n in the decision making process of large decentralized systems, has been widely covered by theoretical and applied research. The problem is particularly apparent in (large) national energy supply systems. Such a system includes: (a) peripheral operating units, and (b) a central planning unit. The peripheral units--which in most European Countries are state owned enterprises--produce, transform and distribute energy. They operate as independent business firms with their own set of criteria--normal business criteria, supplemented by energy related and social criteria [19, 20]. If these units are left alone, to produce their own optimal plans, it is certain that all these 'small' plans put together will constitute a global plan which is suboptimal for the energy system as a whole. This is why a co-ordinating unit--the central planning unit e.g. a Ministry (or Department) of Energy--is needed. The latter is responsible for determining the national energy policy and the optimal structure THE
tered
437
of the energy system; the central unit must make sure that the optimal plan produced by each of the peripheral units is a component of the overall optimal plan of the energy system determined by the central unit. How can this be accomplished? There are two ways of operating: (a) Centralization. The central unit runs its own model, which can be a large LP model. This model is called a "centralized' model throughout this paper. The output of the model gives the overall optimal plan. The central unit issues orders to each peripheral unit, based on the model output. (b) Decentralization. Each peripheral unit runs its own 'peripheral' model. All the outputs of the peripheral models put together can make up a plan which coincides with the overall optimal plan, given by the centralized model, provided that each peripheral model is based on the appropriate input which the central unit sends to the peripheral unit. To produce the 'appropriate' inputs the central unit needs its own small model, known as the 'master' model. This second model, that of decentralized decision making, modelled through the decom-
Samoui/idis, Arabatzi-Ladia--Decentrali-ed Decision Making in the Energy Sector
438
ANALYSIS OF THE N A T I O N A L ENERGY SYSTEM
position principle of LP, is described in this paper. The paper presents a modelling approach which can be used in order to set up a decentralized planning system in the energy supply
Organizational structure The Greek energy system is represented in Fig. 1 as a hierarchical multilevel system. The Ministry of Energy and Natural Resources at the highest level is the central unit, which supervises and co-ordinates the peripheral units. These latter are state owned enterprises that produce and distribute final energy forms (oil product, electricity etc); their status and structure vary greatly [17].
sector.
We start with a large LP model which describes the optimal development of the national energy system structure, based on the unrealistic assumption of an absolutely centralized decision making process. This is a reference model. Taking into account the organizational and decision making structure of the energy sector, we decompose this model into smaller models; thus a model is formulated for each decision centre, i.e. for each peripheral operating unit, and the central unit. We then determine, through a number of successive iterations, the exchange of information required between the central and peripheral units so that the optimal plan of each peripheral unit is a component of the overall optimal. The whole process is illustrated through a case study referring to the national energy system of Greece.
KYSYM
-.- . . . . .
Minrsfry
of
(a)
Public Power Corporation (PPC). Established in 1950; responsible for the construction and operation of hydro and thermal power stations and the establishment of a national network for the transmission and distribution of the produced electrical energy. It is wholly owned by the state and operates for the public interest, under market-economy rules.
(b)
Public
Oil Corporation
. . . . .
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Information
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Fig. I. Structure of the Greek energy system.
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(POC).
Omega. VoL 12, .Vo. 5
Established in 1975: responsible for prospecting, drilling, development and exploitation of oil and gas fields. It operates for the public interest under marketeconomy rules. The state is its sole shareholder and it is represented at its General Assembly by the Ministers of National Economy, Finance and Energy. (c)
.4spropyrgos State Oil Refineries S.A. (ASOR). Established in 1958; responsible for the procurement and refinement of crude oil. The state is its sole shareholder and it is represented at its General Assembly by the Ministers of National Economy and Energy.
Greece has significant lignite and peat deposits. Lignite. used mainly for electricity generation, is mined at present and the PPC is wholly responsible for its exploitation. A new organization for solid fuels may be established, if it is decided to import coal, either for electricity generation or for other industrial uses. The same may happen for gas, if it is decided to import natural gas. Consequently, a fourth group of organizations, responsible for solid and gas fuels, should be included in a representation of the future structure of the Greek energy' system. The PPC is also responsible for the development of alternative energy sources--solar, wind, geothermal--since their main use is for electricity generation. The institutions described above, the central A n a c r o n y m o f the " G o v e r n m e n t C o u n c i l ' in G r e e k , i.e. the M i n i s t e r s o f the N a t i o n a l E c o n o m y , F i n a n c e . Public W o r k s a n d P r e s i d e n c y o f the G o v e r n m e n t .
439
planning unit and the peripheral operating units, are decision centres within the energy system. But they are not the only ones; in addition the millions of energy consumers determine the demand for energy through their decisions. However, in this paper we address ourselves only to the supply' side of the energy' system.
Decision making structure The existing decision-making process on the production side of the energy sector will be analysed. An analysis of the decisions at the consumption end, which is beyond the scope of this paper, is given in [21]. The main energy policy issues are listed in Table 1. The organizations responsible for analysing the various alternative options and making proposals, taking decisions and implementation, are included in the Table. In some of the issues, organizations at a higher system level are involved in the process. They are represented by "KYSYM', t a council which is responsible for co-ordinating the policies of all Government economic departments. The flow of information, proposals and commands between all the organizations in the energy system is shown in Fig. 2. The sector units formulate their short- and medium-term plans, based on the future needs of the economy' for final forms of energy, on technological and economic developments and on general guidelines from the Ministry of Energy'. The plans contain proposals for all the listed issues, except for the stock/reserve levels that are determined exogenously. The Ministry of Energy' is also actively' involved in the planning process for
Table 1. Energy policy issues and responsibilities No [ 2 3 4 5 6 7 8 9 10 II 12 13 14
Issue Energy imports Indigenous energy resource prospecting Exploitation rate of indigenous resources Future structure of energy supply Investment financing Energy R & D expenditure Environmental poiution Energy product prices and tariffs Future structure of energy demand Energy plant operation Material supplies Company organisation and operation Energy conservation Stock'reserve levels
I--KYSYM. 2--Ministry of Energy. 3--PPC, POC, ASOR.
Proposal
Decision
hnplementation
3.2 3 3 3, 2 3.2 3 3.2 3, 2 3 3 3 3.2 3, _.2 2
1 2 2
3 3 3 3.2 3.2 3 3.2 3
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3 3 3.2 3.2 3,2
440
Samouilidis. Arabatzt-Ladia--Decentraliced Decision Making in the EnerTv Sector Socioeconomic
Poricy
r KYSYM ~"i Comm,rtee of Econ Min
I _ _ . ~ i
i
4 T / p \ \
-
j
I
Energy Policy
I
MinPstry of Energy
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I "1
3
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I
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Energy Technology
--,,.- Information / Proposals Level
Organizations
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\ I I
Proposal
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KYSYM
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2
Mm~stry of Energy and Natural Resources Energy State Owned Enterprises
Decismn Making Centre
Implementation Centre Fig. 2. Decision making process in the Greek energy system.
certain important issues that may have longterm consequences, like the future structure of the system, investment financing, tariffand pricing policy, energy conservation and environmental consequences. Depending on the issue, decisions can be made at any of the three system levels. The sector units can decide only on the operating schedule of their plant and on the supply of materials, the rest of the issues are decided at a higher level. The Ministry of Energy decides on the rate of prospecting for and exploitation of indigenous resources, the level of R&D expenditure, the demand forecasts to be adopted and the stock/reserve levels; KYSYM decides on the rest. It should be noted that evaluation of alternative options and the final decision is not made according to a strict, systematic process; rather it is the result of empirical intuitive
weighting of economic, environmental, social and political criteria. The following two observations can be deduced from the analysis above. - - T h e Greek energy system is rather centralized, since all relevant information is gathered and the most important decisions are made at the highest level, - - T h e absence of a unit responsible for global energy policy making is evident. Proposals are made independently by the peripheral units and the directorates within the Energy Ministry, and reach the decision-making centre without going through a process of co-ordination and synthesis. Consequently, fundamental energy policy goals--like energy conservation, substitution of certain final energy forms, the long-term global opti-
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mization of the energy sector in connection with overall economic planning, are not given adequate consideration, at least at the planning stage. THE MODEL The centralized model The exercise starts with a centralized model, which is based on the assumption that there is a one-decision-centre energy system. This particular model is a simplified version of the Greek Energy System Optimization Model- 1 (GRESOM-I). G R E S O M is a series of LP models describing the structure and operation of the national energy system, developed at the National Technical University of Athens, for the quantitative evaluation of major energy policy decisions. A detailed presentation of the first model in the series. G R E S O M - I , is given in [18], and a concise presentation of the set is given in [22]. The model variables are distinguished by:
--Strategic variables, that represent plant capacity for extraction, conversion, transportation and storage of energy. - - Tactical variables, that represent the flows of the various energy forms. The following constraints can be identified:
--Energy demand constraints, specifying that consumers' satisfied. i C(II).X(II)+
demand
for energy must be
C(12).X(12)+ C(I3).X(13)
441
- - Capacity constraints, specifying that installed plant capacity must be adequate to provide the required energy flows. - - T e c h n o l o ~ c a l constraints, related to the technical characteristics of the system. - - O t h e r exogenous constraints, due to economic (i.e. foreign exchange availability), physical (i.e. crude oil or lignite reserves) or political (i.e. nuclear plant installation) factors. The optimization criterion is the minimization of the total discounted cost of satisfying the energy needs of the Greek economy. This cost includes investment, operation and maintenance, energy resource, transmission and distribution cost. The model covers the period 1980 to 2000, divided into three subperiods (1981 to 1985, 1986 to 1990, 1991 to 2000). The following modifications have been introduced to GRESOM-1 (a) The number of periods was reduced from 5 to 3; (b) some new technologies, judged very expensive, were eliminated from the list of candidate technologies; (c) the number of zones, in the (electricity) load duration curve, was reduced from 5 to 2. The modified model, G R E S O M - I a , serves for our purposes as a reference model, through which the global optimum of the national energy system is determined. The Greek energy system is a hierarchical multilevel system with centralized management and all the mathematical models developed for its analytical representation have a blockangular structure, as follows: = Z Objective function
[ A (K,I I ). X(I I ) + A (K, I2), X(I2) + A (K,13).X (•3)
< A ( K ) coupling constraints constrainst of: < B l(M 1) electricity subsystem
B(MI,I1).X(II) B(M2,I2). X(12)
< B2(M2) liquid fuel subsystem B(M3,13).X(I3)
< B3(M3) solid and gas subsystem
where: 11, I2, I3: K: Af 1. M2, M3: C(II), C(12), C(13): X(I 1), X(12), X(13): ,4 (K,I I), A (K, I2). A (K,13): A (K): B ( M I,I l), B(M2,I2), B(M3,I3): B I(MI), B2(M2). B3(M3):
Activity indices for the electricity, oil, solid and gas subsystems respectively. Row index for the coupling constraints. Row indices for the constraints of the subsystems. Unit cost vectors of the activities of the electricity, liquid, solid and gas subsystems respectively. VectOrs of variables for the electricity, liquid, solid and gas subsystems respectively. Coefficient matrices of the coupling constraints for the activities corresponding to the electricity, liquid, solid and gas subsystems respectively. Right hand side coefficients of the coupling constraints. Coefficient matrices of the constraints of the electricity, liquid, solid and gas subsystems respectively. Right hand side coefficients of the constraints of the above systems.
~2
Samouilidis. Arabatzi-Ladia--Decentrali_'ed Decision Making in the Ener~) Sector Table 2. Structure of the GRESOM-ID
Submodel
Subsystem
Organisations
Ac:ivities
GRESOM-ID0
Master Problem Energy Consumption (industrial, transport, domestic)
Ministr? of Energy consumers
GRESOM-IDI
Electricity
PPC
GRESOM-ID2
Oil and products
POC. ASOR
GRESOM-ID3
Lignite and natural gas
SGFC
Energ? Policy making. Energ~ consumption. substitution Production. transmission dis'ribution Production. import. refining, stocking. distribution Production. conversion. import, distribution.
The decentralized model The structure of the above model prompted the idea to use the Decomposition Principle (DP) 2 in developing a model for the representation of the organizational structure of the system [5, 14]. Many applications of the DP to decision making in multilevel systems have been reported in the literature, concerning organizations of different size. For indicative purposes, the studies on a Swedish firm in the paper industry [15], the Danish agricultural sector [4] and economic planning for Hungary [9, 11] are mentioned. Following the conventional wisdom of the DP or in other terms, of decentralization, we do not favour the idea of having an all powerful Minister, sitting at the Ministry of Energy, who determines by decrees the evolution of the system. Instead, we would like to have independent decision centres, one for each organizational unit of Fig. 2, which determine their optimal plans. Our problem is to determine the exchange of information required among the central unit and the peripheral units, so that all the optimal plans put together constitute the overall optimal plan coinciding with that provided by the reference model. We thus arrive at decomposing the centralized model into a number of decentralized models. The G R E S O M - I Centralised Model, is split up into four decentralized models, as shown in Table 2: (1) G R E S O M - 1 D 0 corresponds to the master problem, i.e. describes the decision situation of the Central Unit, which is represented by the Ministry of Energy. (2) GRESOM-1D1 refers to the Electricity subsystem, describing the decision situation of '-For a brief discussion about DP, see Appendix I.
the Public Power Corporation; it covers the production, transmission and distribution of electricity to final consumers. (3) G R E S O M - I D 2 represents the oil and oil products subsystem; thus this model describes the decision situation of two organizational units, the Public Oil Corporation and the Aspropyrgos State Oil Refineries. (4) G R E S O M - I D 3 describes the structure of the solid fuel and natural gas subsectors; there is no organizational unit covering these subsectors, however the presence of this submodel is required for obvious reasons. The overall set of Models is named GRESOM-ID. Representation of the energy consumption subsystem in the same way as the production subsystems would be wrong, since the Ministry's relations with the consumers are significantly different from its relations with the producers, as was mentioned earlier in this paper. It was thought appropriate at this stage of the model development to include a consumption subsystem into the Master problem. This representation is in agreement with the requirements of the DP. M O D E L A P P L I C A T I O N A N D RESULTS
The optimization The algorithm of Dantzig-Wolfe (D-W) which is used here is fully described in the literature [6, 7]. A brief presentation is given in Appendix II. A number of computer programs are used to solve the D - W algorithm. The main program is
Omen.a, Vol. 12, No. 5
LINP [10]. an LP program that exists in the University computer library" and is used to solve the master problem and the subproblems. Two auxiliary programs have been developed especially for this application: (a)
Program DECOI calculates the coefficients of the objective functions of the subproblems at each iteration, according to the relation: R(/L) =
C ( I L ) - y. {A (K, IL).FI(K)}
(4)
K for L = 1 , 2 , 3
where:
R(IL): coefficient of the objective function of the subproblem L, at the end of an iteration. H(K): vector of dual variables of the restricted master problem after an iteration.
443
L (K,J): Coefficient of the j t h variable in the Kth corporate constraint. Data for program DECO2 are modified at each iteration by adding the column-solution of each subproblem to the corresponding matrix of solutions. At each iteration, program DECO2 runs only once, while program DECOI runs once for each subproblem and the columnsolution is introduced into the restricted master problem. A schematic representation of its various steps, as applied to the energy' system, is shown in Fig. 3.
Some numerical results
The numerical computations are in a preliminary stage. So far, we have made a few attempts to arrive at an optimal solution using different starting strategies. First, we initiate the iteration process by calculating some trial solutions of the subproblems, using the following technique: the overall problem was solved, by Vector I-I(K) is the only part of the datum for the Simplex algorithm and the optimal values of program DECOI that is modified at each iter- the dual variables were calculated. Then, five ation. trial solutions for each subproblem were calculated by considering initial values for the dual (b) Program DECO2 calculates the coeffivariables equal to 0.95, 0.975, 1.025, 1.2 and 1.5 cients of the objective function and of the of the optimal. These solutions shaped the first corporate constraints of the restricted restricted master problem, but the application master problem at each iteration, accordof the Simplex algorithm resulted ira an ining to the relations: feasible solution. Therefore, the above technique was considered inappropriate, since our IV{J) = '~ C(J),YP(J) (5) P purpose was to arrive at an acceptable--if not optimal--solution in early iterations. An alterL (K, J) = ~ A (K, J ) - X P ( J ) (6) P native technique was then employed: knowing the optimal solution of the overall problem, it where: was not difficult to find an arbitrary feasible W(J): coefficient of t h e j t h variable in the solution, which initiated the D - W algorithm. objective function of the restricted 'masThe master problem was then solved and the ter" problem. dual values, corresponding to the prices of the X'(J): matrix of solutions of the subcommon resources, were calculated. problem that have been introduced into These prices are transmitted to the subthe restricted 'master' problem up to the problems through the modified coefficients of pth iteration. their objective functions (calculated by DECOI ). Table 3. Application of the D--W algorithm to G R E S O M - D I Next, the subproblems are solved and their optimal proposals are passed to the restricted Optimun value of the objective function lnteration (in Drachmas. 100 drs = USS I). master problem, through the coefficients of its objective function and coupling constraints (calI 1792570663, 2 1792570663. culated by DECO2) The second iteration starts 3 1792570663. with the solution of the new restricted master 4 1792570663. 5 1789875345. problem. The results for the first six iterations 6 [789752663. are shown in Table 3. There is only a small
444
Samouilidis. Arabatzi-Ladia--Decentrali:ed
Decision M.king in the Energy Sector
DECO '~
/
SuOproblem
Test
3
] Solu,,oo
Solution
o, S,'b 3
.!
problem
Fig. 3. Computer software support for the D-W algorithm.
change in the value of the objective function, which in the sixth iteration is still 16~°/ greater than the optimal. Different starting strategies are now tested, aiming at a higher rate of convergence in early iterations.
Organizational interpretation A very sketchy organizational interpretation of the modelling approach was given in the introductory section of this paper. A detailed presentation follows. The central unit, corresponding to the master problem, co-ordinates the peripheral units through the mechanism of pricing of common resources. Three types of common resources can be distinguished. (i) Indigenous non-renewable energy resources (lignite and crude oil). The available resources of this type are determined exogenously and change only when new discoveries are made. In G R E S O M - I D these resources are: (a) lignite, used in electricity generation and in the industrial and domestic sectors (solid and gas fuels subsystem). (b) crude oil, used--after distillation, cracking, etc.--in electricity generation and as a combustible in all consumption sectors.
The energy production subsystems utilize these common resources in quantities that optimize their plans, according to prices set by the central unit. The corresponding coupling constraints in the master problem are four for lignite and crude oil available reserves, and three for lignite balances. (ii) hnported resources (oil prod,lcts, coal attd plant equipment). Available resources are not determined exogenously, in this case, and can be unlimited. However, their availability is bounded by the central unit, since their cost is a significant part of the total energy system cost a n d - - m o r e importantly--has to be paid in foreign exchange. Imported oil products are used in the electricity and liquid fuels subsystems, while plant equipment is used in all three subsystems. There are six coupling constraints in the master problem for fuel and diesel oil balances and one for foreign exchange; this latter includes all energy related imports (i.e. oil, coal, machinery and equipment for the energy sector). (iii) Useful energy--Energy services. Although 'useful energy' is not a 'resource' in strict terms, it serves some purpose to be considered as such. The subproblems correspond to the
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peripheral units that optimize their o p e r a t i o n according to their own criteria. Thus, they aim to satisfy the requirements of the e c o n o m y in useful energy a n d to maximize their profit, which is a function of: (a) the prices of final energy, (b) the prices of energy resources, used as inputs by each peripheral unit. Both prices are set by the central unit, so that each peripheral unit supplies the optimal share of the energy market. There are nine coupling c o n s t r a i n t s in the "master' p r o b l e m which represent the satisfaction of useful energy d e m a n d . CONCLUDING
REMARKS
In this paper we investigate the possibility of using the DP, a n d the D - W algorithm in particular, as a tool in the design o f a decentralized d e c i s i o n - m a k i n g process for the Energy System. Specifically the proposed method, (a) enables a check on the compatibility of the p r o g r a m m e s , s u b m i t t e d by the peripheral units to the central unit, with the overall ' o p t i m u m ' p r o g r a m of the energy system, a n d (b) can be used as a reference framework for interaction a n d n e g o t i a t i o n s between the central a n d the peripheral units, in m e d i u m - and long-term p l a n n i n g , for the allocation of funds within the n a t i o n a l energy i n v e s t m e n t budget. However, as pointed out in the i n t r o d u c t o r y p a r a g r a p h of this paper, we have limited ourselves to the supply side of the energy system. The next step in our research is to extend the modelling a p p r o a c h in order to cover the d e m a n d side. We shall include in the set of interacting models d e c i s i o n - m a k i n g models that corres p o n d to the energy consumers. The central unit has to intervene using the price m e c h a n i s m , in order to make sure that c o n s u m e r s ' choices are tuned to the overall optimal structure of the energy system.
REFERENCES
..t45
4. Christensen J and Obel B (1978) Simulation of decentralized planning in two Danish organizations using linear programming decomposition. 5I~rnt Sci. 15, 1658-[667. 5. Dantzig GB ( 19631Linear Programming and Extensions. Princeton University Press. New Jersey. 6. Dantzig G and Wolfe P (1960) Decomposition principles for linear programs. Ops Res. 8, 101 Ill. 7. Dantzig G and Wolfe P (1961) The decomposition algorithm for linear programming. Econometrica 89{4). 767-778. 8. Dirickx Y and Jennergren P (19791 5~vstems Anah'sis h)" Multilevel Methods: with Applications to Economies and Management. IIASA, Wiley, New York. 9. Goreux LM and Manne AS (1973) (Ed.) Multilecel Planning, Case Studies in Mexico. North-Holland,
Amsterdam. 10. Kornai J and Liptak TH (1965) Two level planning. Econometrica 33( [ ), 141-169. l l. Kornai J (1969) Man-machine planning. Econ. Phmn. 9(3) 204-234. 12. Land AH and Powell S (1973) Fortran Codes fi~r Mathematical Progratnming: Linear, Qua&'atic and Discrete. Wile,,', New York. 13. Lasdon LS (1970) Optimi.zation Theor.v j'or Large Systems. Macmillan, New York. 14. Obel B (1981) Issues of Organisational Design. Chaps
l-3. Pergamon Press, Oxford. 15. Obel B (1981) Issues of Organisational Design. pp. 197-200. Pergamon Press, Oxford. 16. Obel B (1981) Issues of Organisational Design. pp. 195-197. Pergamon Press, Oxford. 17. Provopoulos G (1982) Public enterprises and organisations. Economic Theory and Greek Practice. Institute of Economic and Industrial Research, Athens. 18. Samouilidis JE and Arabatzi-Ladia A (1982) A Model for the Greek energy system. Eur. J. Opl Res. 9, 144-160. 19. Samouilidis JE (1982) A systemic approach to energy policy. In Proceedings of the First National Congress for Soft Energy Resources. Institute of Solar Technology, Thessaloniki, Greece, October 20-22, pp. 29-45. (In Greek). 20. Samouilidis JE (1982) Systems analysis for policy formulation. Paper presented at the Joint Congress of the Hellenic Operational Research Society attd the Greek Management Association, Delphi, Greece. (In Greek).
21. Samouilidis JE, Berahas SA and Psarras JE (1983) Centralized versus decentralized decision making. Energy Poli
ADDRESS [--OR CORRESPONDENCE:
APPENDIX
I
The decomposition principle
1. Atkins D (1974) Managerial decentralization and decomposition in mathematical programming. Opl Res. Q. 25, 615-624. 2. Burton RM and Obel B (1977)The multilevel approach to organisational issues of the firm--A critical review. Omega 5(4), 395-414. 3. Burton RM and Obel B (1980) The efficiency of the price, budget and mixed approaches under varying a priori information levels for decentralized planning. Mgmt Sci. 4, 401-417.
Application of the linear programming method to larger and more complex problems, in connection with the limited capabilities of computers in the early sixties, introduced the use of the DP in the development of special algorithms, adapted to the characteristics of each particular type of problem [4]. It was soon recognized that the DP apart from helping to overcome computational difficulties,could also be employed in the development of models for decentralized planning in multilevel organizational systems.
Samouilidis, .4rubutzi-Ladia--Decentrali-ed
446
The t'ollov, ing example shov, s a simplified representation of the problem in a two-level system, frequently met in the literature. max CI .V~ + C: .V:... Cp,Vp
Decision Making in the Energy Sector subject
to
alex I + a~: X, + aI3 fl + at-t Y: + a:5 Y; <2 C
I
a:t ,~(i + az2 ,'(2 <-- C,
(1)
ajt.,~'~-'-a3.-X 2 N C 3
subject to At .t'l + A,.X: . . . . 4pXe = <_bo B~ .V~ = < b I
B , X , = <_b2
B.,V~ =
a~, YI + a u I"., + a45 Y3 < C~
(2~
(3)
<_be
A7 > 0 . i = 1 . . . . p where:
Ci, A'i, bi are vectors Ai. Bi are matrices It can be seen that the problem has a block-angular structure appropriate for decomposition and can be divided into ( p + l) subproblems. Each one of p subproblems corresponds to the activities f i of subsystem i. and the remaining subproblem, called the master problem, corresponds to the Central Unit of the system and co-ordinates the solutions of the p subproblems so that coupling constraints (2) are satisfied. The main difficulty in solving this type of problem is the co-ordination between the master problem and the subproblems. Algorithmically, this means the sending up and down of values of selected parameters between the master problem and the subproblems. Organizationally co-cordination means a two-way transfer of information and c o m m a n d s . There are two main approaches to the problem. The pricing approach whereby the information passed by the master problem to the subproblems concerns prices--the Dantzig-Wolfe algorithm [5] i,; the best-known representative in this area. The budgeting approach whereby the information sent down concerns resource allocations--the Kornai-Liptak algorithm [101 is representative of this area. Comprehensive reviews of the literature on the subject are given in [2, 3]. The objective of the problem is to find a feasible and, if possible, o p t i m u m solution. An important criterion in evaluating an algorithm is the rate of convergence during the first interactions, and its importance is enhanced in applications concerning decentralized planning in multilevel organizations. Research results have shown that the rate of convergence depends on the organizational structure of the system and in particular on:
where the P ' s are profit coefficients, a,j's are input/output coefficients, C's are fixed resources, and X's represent decision variables for subunit 1 and Y's for subunit 2. It is the special structure of this problem which lends itself to the decomposition and the analogous hierarchical model of the sector. First. note that the objective function (for profit) is linear, and therefore separable, and thus the portion with the X's can be assigned to subunit I and the Y's to subunit 2 without interference (or externalities). Next, the first constraint affects all decision variables of the sector and is called the c o m m o n constraint. The next two constraints involve only the X's and the last constraint only the Y's. These are called subunit constraints. Similarly. C~ is called a c o m m o n resource, and C,, C 3. Ca subunit resources. The organizational algorithm consists of the executive, or co-ordination unit, solving a problem and then passing information to the subunits. The subunits, in turn, solve a planning problem and send information back to the executive. For a linear programming model, each submodel is also a linear program for the Dantzig-Wolfe approach terminating at an overall optimal solution after a finite number of iterations. This process is given in more detail: let us begin with any subunit problem at an arbitrary iteration, e.g. subunit 1 is asked to solve the following LP: max R~ .V~ + R: ,g., with respect to
A"I, X~.
a21X ~+ a2,,.X: <, C, a3zX I -- a32X 2 <_.C 3 where R~ and R 2 are modified profit co-efficients which are communicated to subunit I by the executive or master program. Subunit I then, sends the optimal values of ,Y~, X2 back to t h e executive unit. The executive program consists of the original objective function and the c o m m o n constraints plus convex combination constraints. The program determines the best convex combination of all past solutions sent up by the subunits. The co-ordination program is: max
- - T h e n u m b e r of coupling contraints. - - T h e n u m b e r of subproblems. - - T h e initial solution.
(2)
subject to
f ~,(P~ X~ + P: X') + v,(I)3Y{ + P, Y',_+ P~ Y~ ) ¢=I
with respect to ]i T,
~ t 1, . . . .
t' I ....
, t"T
subject to APPENDIX
T
II
7. ~z,(at. X'~ + at,X',_) + c,(a,3 g'~ + at~ Y', + a~5 g~) < C~ (3)
The Dantzig-Wolfe algorithm Consider that the energy sector has a two-level structure, i.e. a central unit (Ministry of Energy) and two peripheral subunits, subunit 1, e.g. Public Power C o m p a n y and subunit 2, e.g. State Oil Refineries. The mathematical model which represents the decision problem is given as follows: max P = Pt X, + Pz'V: + P~ Yt + P~ Y'- + P~ Y~
1"
~ ,u,= 1 t=l r t=l
#~, t'~ > 0 Vt
with respect to
Yl, X:. YI, Y'., r'3
t=l
(1)
The decision variables are the linear weightings lL's and o's. The solutions to the organizational decision problem ,g"
Orneea, Vol. IZ No. 5 and YJ are deri',ed from the master problem. The? are v*ritten fully as: .Y~ = .a,..F,I - . . . + #,.'(,r and I"~-= t', F) - : - . . . ,
t': F,r
where the superscript 7" refers to the optimal solution. The master primal p r o g r a m is not of great interest. The interesting information is in its dual. The dual variabte for the c o m m o n constraint, call it =t for C~, measures the marginal value, an o p p o r t u n i t y cost of that c o m m o n resource. Subunits which use this resource are then "charged' for its use by substracting the "cost" of the resource from their profit coefficient. The modified profit
447
coefficient for )(k is p v e n as: R~ = R I - - a , l ~ l . T h a t is, the net profit coefficient for ,~'~ is the original coefficient minus the cost of the use of the scarce c o m m o n resource. Similarly, modified profit coefficients are calculated for X, and Y's. Briefly, the executive sends modified prices (hence the term pricing approach) to the subunits, the subunits calculate their best solution for outputs, and send these back to the executive who in turn determines the value of the scarce corporate resources, and adjusts the prices. The executive provides a solution for prices, the subunits t'or outputs.
0305-0483 8.-t8 3 . 0 0 - 0 . 0 0 Copyright ~ 1984 Pergamon Press Ltd
Printed in Great Britain. All rights reser~,ed
Modelling Decentralized Decision Making in the Energy Sector J - E M M A N U E L SAMOUILIDIS A ARABATZI-LADIA Energy Policy Unit, National Technical University of Athens, Greece (Received November 1983: in revised form January 1984) Large linear programming models, which have been widely used to determine the optimal structure of national energy systems, are based on the assumption that there is an absolutely centralized decision making process within the energy system. In this paper an attempt is made to match the real decision structure of a given energy system, by decomposing an LP energy model into smaller models, with the corresponding system decision centres. This is done by applying the 'transfer price' algorithm of Dantzing and Wolfe. The 'master' problem corresponds to the central planning unit, i.e. a Ministry of Energy, whereas the subproblems correspond to peripheral operating units, i.e. enterprises, usually state owned, which produce and distribute the energy carriers. The optimal plans of the peripheral units are submitted to the central unit, which through the mechanism of pricing of both common resources, inputs and energy services outputs, co-ordinates the overall planning of the energy system. An illustrative example is given referring to the Hellenic national energy system. The research reported is placed within a wider research endeavour, whose objectives and main line of work are also given.
INTRODUCTION PROBLEM O f suboptimization, e n c o u n in the decision making process of large decentralized systems, has been widely covered by theoretical and applied research. The problem is particularly apparent in (large) national energy supply systems. Such a system includes: (a) peripheral operating units, and (b) a central planning unit. The peripheral units--which in most European Countries are state owned enterprises--produce, transform and distribute energy. They operate as independent business firms with their own set of criteria--normal business criteria, supplemented by energy related and social criteria [19, 20]. If these units are left alone, to produce their own optimal plans, it is certain that all these 'small' plans put together will constitute a global plan which is suboptimal for the energy system as a whole. This is why a co-ordinating unit--the central planning unit e.g. a Ministry (or Department) of Energy--is needed. The latter is responsible for determining the national energy policy and the optimal structure THE
tered
437
of the energy system; the central unit must make sure that the optimal plan produced by each of the peripheral units is a component of the overall optimal plan of the energy system determined by the central unit. How can this be accomplished? There are two ways of operating: (a) Centralization. The central unit runs its own model, which can be a large LP model. This model is called a "centralized' model throughout this paper. The output of the model gives the overall optimal plan. The central unit issues orders to each peripheral unit, based on the model output. (b) Decentralization. Each peripheral unit runs its own 'peripheral' model. All the outputs of the peripheral models put together can make up a plan which coincides with the overall optimal plan, given by the centralized model, provided that each peripheral model is based on the appropriate input which the central unit sends to the peripheral unit. To produce the 'appropriate' inputs the central unit needs its own small model, known as the 'master' model. This second model, that of decentralized decision making, modelled through the decom-
Samoui/idis, Arabatzi-Ladia--Decentrali-ed Decision Making in the Energy Sector
438
ANALYSIS OF THE N A T I O N A L ENERGY SYSTEM
position principle of LP, is described in this paper. The paper presents a modelling approach which can be used in order to set up a decentralized planning system in the energy supply
Organizational structure The Greek energy system is represented in Fig. 1 as a hierarchical multilevel system. The Ministry of Energy and Natural Resources at the highest level is the central unit, which supervises and co-ordinates the peripheral units. These latter are state owned enterprises that produce and distribute final energy forms (oil product, electricity etc); their status and structure vary greatly [17].
sector.
We start with a large LP model which describes the optimal development of the national energy system structure, based on the unrealistic assumption of an absolutely centralized decision making process. This is a reference model. Taking into account the organizational and decision making structure of the energy sector, we decompose this model into smaller models; thus a model is formulated for each decision centre, i.e. for each peripheral operating unit, and the central unit. We then determine, through a number of successive iterations, the exchange of information required between the central and peripheral units so that the optimal plan of each peripheral unit is a component of the overall optimal. The whole process is illustrated through a case study referring to the national energy system of Greece.
KYSYM
-.- . . . . .
Minrsfry
of
(a)
Public Power Corporation (PPC). Established in 1950; responsible for the construction and operation of hydro and thermal power stations and the establishment of a national network for the transmission and distribution of the produced electrical energy. It is wholly owned by the state and operates for the public interest, under market-economy rules.
(b)
Public
Oil Corporation
. . . . .
Influences
Energy
I
-- -- Other
Mimstries
I I I
I I
t
I I
[
I
l
Other
S.A.
1
i,
c °
g
o
t.-
..J
Energy
Command . . . .
Energy
Producers
--~
Information
/
D~recf
Consumers
Control
/ Indtrecr
Control
Fig. I. Structure of the Greek energy system.
£3
(POC).
Omega. VoL 12, .Vo. 5
Established in 1975: responsible for prospecting, drilling, development and exploitation of oil and gas fields. It operates for the public interest under marketeconomy rules. The state is its sole shareholder and it is represented at its General Assembly by the Ministers of National Economy, Finance and Energy. (c)
.4spropyrgos State Oil Refineries S.A. (ASOR). Established in 1958; responsible for the procurement and refinement of crude oil. The state is its sole shareholder and it is represented at its General Assembly by the Ministers of National Economy and Energy.
Greece has significant lignite and peat deposits. Lignite. used mainly for electricity generation, is mined at present and the PPC is wholly responsible for its exploitation. A new organization for solid fuels may be established, if it is decided to import coal, either for electricity generation or for other industrial uses. The same may happen for gas, if it is decided to import natural gas. Consequently, a fourth group of organizations, responsible for solid and gas fuels, should be included in a representation of the future structure of the Greek energy' system. The PPC is also responsible for the development of alternative energy sources--solar, wind, geothermal--since their main use is for electricity generation. The institutions described above, the central A n a c r o n y m o f the " G o v e r n m e n t C o u n c i l ' in G r e e k , i.e. the M i n i s t e r s o f the N a t i o n a l E c o n o m y , F i n a n c e . Public W o r k s a n d P r e s i d e n c y o f the G o v e r n m e n t .
439
planning unit and the peripheral operating units, are decision centres within the energy system. But they are not the only ones; in addition the millions of energy consumers determine the demand for energy through their decisions. However, in this paper we address ourselves only to the supply' side of the energy' system.
Decision making structure The existing decision-making process on the production side of the energy sector will be analysed. An analysis of the decisions at the consumption end, which is beyond the scope of this paper, is given in [21]. The main energy policy issues are listed in Table 1. The organizations responsible for analysing the various alternative options and making proposals, taking decisions and implementation, are included in the Table. In some of the issues, organizations at a higher system level are involved in the process. They are represented by "KYSYM', t a council which is responsible for co-ordinating the policies of all Government economic departments. The flow of information, proposals and commands between all the organizations in the energy system is shown in Fig. 2. The sector units formulate their short- and medium-term plans, based on the future needs of the economy' for final forms of energy, on technological and economic developments and on general guidelines from the Ministry of Energy'. The plans contain proposals for all the listed issues, except for the stock/reserve levels that are determined exogenously. The Ministry of Energy' is also actively' involved in the planning process for
Table 1. Energy policy issues and responsibilities No [ 2 3 4 5 6 7 8 9 10 II 12 13 14
Issue Energy imports Indigenous energy resource prospecting Exploitation rate of indigenous resources Future structure of energy supply Investment financing Energy R & D expenditure Environmental poiution Energy product prices and tariffs Future structure of energy demand Energy plant operation Material supplies Company organisation and operation Energy conservation Stock'reserve levels
I--KYSYM. 2--Ministry of Energy. 3--PPC, POC, ASOR.
Proposal
Decision
hnplementation
3.2 3 3 3, 2 3.2 3 3.2 3, 2 3 3 3 3.2 3, _.2 2
1 2 2
3 3 3 3.2 3.2 3 3.2 3
~ _, ~ .~ 2. 2,
3 3 3.2 3.2 3,2
440
Samouilidis. Arabatzt-Ladia--Decentraliced Decision Making in the EnerTv Sector Socioeconomic
Poricy
r KYSYM ~"i Comm,rtee of Econ Min
I _ _ . ~ i
i
4 T / p \ \
-
j
I
Energy Policy
I
MinPstry of Energy
f I l/
I "1
3
J
I
/
t
3
1
/
\
I, :3 J
i i
t
I
I
Energy Technology
--,,.- Information / Proposals Level
Organizations
,-. Commands / I \
r L__
\ I I
Proposal
I
KYSYM
Decision
2
Mm~stry of Energy and Natural Resources Energy State Owned Enterprises
Decismn Making Centre
Implementation Centre Fig. 2. Decision making process in the Greek energy system.
certain important issues that may have longterm consequences, like the future structure of the system, investment financing, tariffand pricing policy, energy conservation and environmental consequences. Depending on the issue, decisions can be made at any of the three system levels. The sector units can decide only on the operating schedule of their plant and on the supply of materials, the rest of the issues are decided at a higher level. The Ministry of Energy decides on the rate of prospecting for and exploitation of indigenous resources, the level of R&D expenditure, the demand forecasts to be adopted and the stock/reserve levels; KYSYM decides on the rest. It should be noted that evaluation of alternative options and the final decision is not made according to a strict, systematic process; rather it is the result of empirical intuitive
weighting of economic, environmental, social and political criteria. The following two observations can be deduced from the analysis above. - - T h e Greek energy system is rather centralized, since all relevant information is gathered and the most important decisions are made at the highest level, - - T h e absence of a unit responsible for global energy policy making is evident. Proposals are made independently by the peripheral units and the directorates within the Energy Ministry, and reach the decision-making centre without going through a process of co-ordination and synthesis. Consequently, fundamental energy policy goals--like energy conservation, substitution of certain final energy forms, the long-term global opti-
Omega, Vol. 12, No. 5
mization of the energy sector in connection with overall economic planning, are not given adequate consideration, at least at the planning stage. THE MODEL The centralized model The exercise starts with a centralized model, which is based on the assumption that there is a one-decision-centre energy system. This particular model is a simplified version of the Greek Energy System Optimization Model- 1 (GRESOM-I). G R E S O M is a series of LP models describing the structure and operation of the national energy system, developed at the National Technical University of Athens, for the quantitative evaluation of major energy policy decisions. A detailed presentation of the first model in the series. G R E S O M - I , is given in [18], and a concise presentation of the set is given in [22]. The model variables are distinguished by:
--Strategic variables, that represent plant capacity for extraction, conversion, transportation and storage of energy. - - Tactical variables, that represent the flows of the various energy forms. The following constraints can be identified:
--Energy demand constraints, specifying that consumers' satisfied. i C(II).X(II)+
demand
for energy must be
C(12).X(12)+ C(I3).X(13)
441
- - Capacity constraints, specifying that installed plant capacity must be adequate to provide the required energy flows. - - T e c h n o l o ~ c a l constraints, related to the technical characteristics of the system. - - O t h e r exogenous constraints, due to economic (i.e. foreign exchange availability), physical (i.e. crude oil or lignite reserves) or political (i.e. nuclear plant installation) factors. The optimization criterion is the minimization of the total discounted cost of satisfying the energy needs of the Greek economy. This cost includes investment, operation and maintenance, energy resource, transmission and distribution cost. The model covers the period 1980 to 2000, divided into three subperiods (1981 to 1985, 1986 to 1990, 1991 to 2000). The following modifications have been introduced to GRESOM-1 (a) The number of periods was reduced from 5 to 3; (b) some new technologies, judged very expensive, were eliminated from the list of candidate technologies; (c) the number of zones, in the (electricity) load duration curve, was reduced from 5 to 2. The modified model, G R E S O M - I a , serves for our purposes as a reference model, through which the global optimum of the national energy system is determined. The Greek energy system is a hierarchical multilevel system with centralized management and all the mathematical models developed for its analytical representation have a blockangular structure, as follows: = Z Objective function
[ A (K,I I ). X(I I ) + A (K, I2), X(I2) + A (K,13).X (•3)
< A ( K ) coupling constraints constrainst of: < B l(M 1) electricity subsystem
B(MI,I1).X(II) B(M2,I2). X(12)
< B2(M2) liquid fuel subsystem B(M3,13).X(I3)
< B3(M3) solid and gas subsystem
where: 11, I2, I3: K: Af 1. M2, M3: C(II), C(12), C(13): X(I 1), X(12), X(13): ,4 (K,I I), A (K, I2). A (K,13): A (K): B ( M I,I l), B(M2,I2), B(M3,I3): B I(MI), B2(M2). B3(M3):
Activity indices for the electricity, oil, solid and gas subsystems respectively. Row index for the coupling constraints. Row indices for the constraints of the subsystems. Unit cost vectors of the activities of the electricity, liquid, solid and gas subsystems respectively. VectOrs of variables for the electricity, liquid, solid and gas subsystems respectively. Coefficient matrices of the coupling constraints for the activities corresponding to the electricity, liquid, solid and gas subsystems respectively. Right hand side coefficients of the coupling constraints. Coefficient matrices of the constraints of the electricity, liquid, solid and gas subsystems respectively. Right hand side coefficients of the constraints of the above systems.
~2
Samouilidis. Arabatzi-Ladia--Decentrali_'ed Decision Making in the Ener~) Sector Table 2. Structure of the GRESOM-ID
Submodel
Subsystem
Organisations
Ac:ivities
GRESOM-ID0
Master Problem Energy Consumption (industrial, transport, domestic)
Ministr? of Energy consumers
GRESOM-IDI
Electricity
PPC
GRESOM-ID2
Oil and products
POC. ASOR
GRESOM-ID3
Lignite and natural gas
SGFC
Energ? Policy making. Energ~ consumption. substitution Production. transmission dis'ribution Production. import. refining, stocking. distribution Production. conversion. import, distribution.
The decentralized model The structure of the above model prompted the idea to use the Decomposition Principle (DP) 2 in developing a model for the representation of the organizational structure of the system [5, 14]. Many applications of the DP to decision making in multilevel systems have been reported in the literature, concerning organizations of different size. For indicative purposes, the studies on a Swedish firm in the paper industry [15], the Danish agricultural sector [4] and economic planning for Hungary [9, 11] are mentioned. Following the conventional wisdom of the DP or in other terms, of decentralization, we do not favour the idea of having an all powerful Minister, sitting at the Ministry of Energy, who determines by decrees the evolution of the system. Instead, we would like to have independent decision centres, one for each organizational unit of Fig. 2, which determine their optimal plans. Our problem is to determine the exchange of information required among the central unit and the peripheral units, so that all the optimal plans put together constitute the overall optimal plan coinciding with that provided by the reference model. We thus arrive at decomposing the centralized model into a number of decentralized models. The G R E S O M - I Centralised Model, is split up into four decentralized models, as shown in Table 2: (1) G R E S O M - 1 D 0 corresponds to the master problem, i.e. describes the decision situation of the Central Unit, which is represented by the Ministry of Energy. (2) GRESOM-1D1 refers to the Electricity subsystem, describing the decision situation of '-For a brief discussion about DP, see Appendix I.
the Public Power Corporation; it covers the production, transmission and distribution of electricity to final consumers. (3) G R E S O M - I D 2 represents the oil and oil products subsystem; thus this model describes the decision situation of two organizational units, the Public Oil Corporation and the Aspropyrgos State Oil Refineries. (4) G R E S O M - I D 3 describes the structure of the solid fuel and natural gas subsectors; there is no organizational unit covering these subsectors, however the presence of this submodel is required for obvious reasons. The overall set of Models is named GRESOM-ID. Representation of the energy consumption subsystem in the same way as the production subsystems would be wrong, since the Ministry's relations with the consumers are significantly different from its relations with the producers, as was mentioned earlier in this paper. It was thought appropriate at this stage of the model development to include a consumption subsystem into the Master problem. This representation is in agreement with the requirements of the DP. M O D E L A P P L I C A T I O N A N D RESULTS
The optimization The algorithm of Dantzig-Wolfe (D-W) which is used here is fully described in the literature [6, 7]. A brief presentation is given in Appendix II. A number of computer programs are used to solve the D - W algorithm. The main program is
Omen.a, Vol. 12, No. 5
LINP [10]. an LP program that exists in the University computer library" and is used to solve the master problem and the subproblems. Two auxiliary programs have been developed especially for this application: (a)
Program DECOI calculates the coefficients of the objective functions of the subproblems at each iteration, according to the relation: R(/L) =
C ( I L ) - y. {A (K, IL).FI(K)}
(4)
K for L = 1 , 2 , 3
where:
R(IL): coefficient of the objective function of the subproblem L, at the end of an iteration. H(K): vector of dual variables of the restricted master problem after an iteration.
443
L (K,J): Coefficient of the j t h variable in the Kth corporate constraint. Data for program DECO2 are modified at each iteration by adding the column-solution of each subproblem to the corresponding matrix of solutions. At each iteration, program DECO2 runs only once, while program DECOI runs once for each subproblem and the columnsolution is introduced into the restricted master problem. A schematic representation of its various steps, as applied to the energy' system, is shown in Fig. 3.
Some numerical results
The numerical computations are in a preliminary stage. So far, we have made a few attempts to arrive at an optimal solution using different starting strategies. First, we initiate the iteration process by calculating some trial solutions of the subproblems, using the following technique: the overall problem was solved, by Vector I-I(K) is the only part of the datum for the Simplex algorithm and the optimal values of program DECOI that is modified at each iter- the dual variables were calculated. Then, five ation. trial solutions for each subproblem were calculated by considering initial values for the dual (b) Program DECO2 calculates the coeffivariables equal to 0.95, 0.975, 1.025, 1.2 and 1.5 cients of the objective function and of the of the optimal. These solutions shaped the first corporate constraints of the restricted restricted master problem, but the application master problem at each iteration, accordof the Simplex algorithm resulted ira an ining to the relations: feasible solution. Therefore, the above technique was considered inappropriate, since our IV{J) = '~ C(J),YP(J) (5) P purpose was to arrive at an acceptable--if not optimal--solution in early iterations. An alterL (K, J) = ~ A (K, J ) - X P ( J ) (6) P native technique was then employed: knowing the optimal solution of the overall problem, it where: was not difficult to find an arbitrary feasible W(J): coefficient of t h e j t h variable in the solution, which initiated the D - W algorithm. objective function of the restricted 'masThe master problem was then solved and the ter" problem. dual values, corresponding to the prices of the X'(J): matrix of solutions of the subcommon resources, were calculated. problem that have been introduced into These prices are transmitted to the subthe restricted 'master' problem up to the problems through the modified coefficients of pth iteration. their objective functions (calculated by DECOI ). Table 3. Application of the D--W algorithm to G R E S O M - D I Next, the subproblems are solved and their optimal proposals are passed to the restricted Optimun value of the objective function lnteration (in Drachmas. 100 drs = USS I). master problem, through the coefficients of its objective function and coupling constraints (calI 1792570663, 2 1792570663. culated by DECO2) The second iteration starts 3 1792570663. with the solution of the new restricted master 4 1792570663. 5 1789875345. problem. The results for the first six iterations 6 [789752663. are shown in Table 3. There is only a small
444
Samouilidis. Arabatzi-Ladia--Decentrali:ed
Decision M.king in the Energy Sector
DECO '~
/
SuOproblem
Test
3
] Solu,,oo
Solution
o, S,'b 3
.!
problem
Fig. 3. Computer software support for the D-W algorithm.
change in the value of the objective function, which in the sixth iteration is still 16~°/ greater than the optimal. Different starting strategies are now tested, aiming at a higher rate of convergence in early iterations.
Organizational interpretation A very sketchy organizational interpretation of the modelling approach was given in the introductory section of this paper. A detailed presentation follows. The central unit, corresponding to the master problem, co-ordinates the peripheral units through the mechanism of pricing of common resources. Three types of common resources can be distinguished. (i) Indigenous non-renewable energy resources (lignite and crude oil). The available resources of this type are determined exogenously and change only when new discoveries are made. In G R E S O M - I D these resources are: (a) lignite, used in electricity generation and in the industrial and domestic sectors (solid and gas fuels subsystem). (b) crude oil, used--after distillation, cracking, etc.--in electricity generation and as a combustible in all consumption sectors.
The energy production subsystems utilize these common resources in quantities that optimize their plans, according to prices set by the central unit. The corresponding coupling constraints in the master problem are four for lignite and crude oil available reserves, and three for lignite balances. (ii) hnported resources (oil prod,lcts, coal attd plant equipment). Available resources are not determined exogenously, in this case, and can be unlimited. However, their availability is bounded by the central unit, since their cost is a significant part of the total energy system cost a n d - - m o r e importantly--has to be paid in foreign exchange. Imported oil products are used in the electricity and liquid fuels subsystems, while plant equipment is used in all three subsystems. There are six coupling constraints in the master problem for fuel and diesel oil balances and one for foreign exchange; this latter includes all energy related imports (i.e. oil, coal, machinery and equipment for the energy sector). (iii) Useful energy--Energy services. Although 'useful energy' is not a 'resource' in strict terms, it serves some purpose to be considered as such. The subproblems correspond to the
Omega, Vol. IZ .Vo. 5
peripheral units that optimize their o p e r a t i o n according to their own criteria. Thus, they aim to satisfy the requirements of the e c o n o m y in useful energy a n d to maximize their profit, which is a function of: (a) the prices of final energy, (b) the prices of energy resources, used as inputs by each peripheral unit. Both prices are set by the central unit, so that each peripheral unit supplies the optimal share of the energy market. There are nine coupling c o n s t r a i n t s in the "master' p r o b l e m which represent the satisfaction of useful energy d e m a n d . CONCLUDING
REMARKS
In this paper we investigate the possibility of using the DP, a n d the D - W algorithm in particular, as a tool in the design o f a decentralized d e c i s i o n - m a k i n g process for the Energy System. Specifically the proposed method, (a) enables a check on the compatibility of the p r o g r a m m e s , s u b m i t t e d by the peripheral units to the central unit, with the overall ' o p t i m u m ' p r o g r a m of the energy system, a n d (b) can be used as a reference framework for interaction a n d n e g o t i a t i o n s between the central a n d the peripheral units, in m e d i u m - and long-term p l a n n i n g , for the allocation of funds within the n a t i o n a l energy i n v e s t m e n t budget. However, as pointed out in the i n t r o d u c t o r y p a r a g r a p h of this paper, we have limited ourselves to the supply side of the energy system. The next step in our research is to extend the modelling a p p r o a c h in order to cover the d e m a n d side. We shall include in the set of interacting models d e c i s i o n - m a k i n g models that corres p o n d to the energy consumers. The central unit has to intervene using the price m e c h a n i s m , in order to make sure that c o n s u m e r s ' choices are tuned to the overall optimal structure of the energy system.
REFERENCES
..t45
4. Christensen J and Obel B (1978) Simulation of decentralized planning in two Danish organizations using linear programming decomposition. 5I~rnt Sci. 15, 1658-[667. 5. Dantzig GB ( 19631Linear Programming and Extensions. Princeton University Press. New Jersey. 6. Dantzig G and Wolfe P (1960) Decomposition principles for linear programs. Ops Res. 8, 101 Ill. 7. Dantzig G and Wolfe P (1961) The decomposition algorithm for linear programming. Econometrica 89{4). 767-778. 8. Dirickx Y and Jennergren P (19791 5~vstems Anah'sis h)" Multilevel Methods: with Applications to Economies and Management. IIASA, Wiley, New York. 9. Goreux LM and Manne AS (1973) (Ed.) Multilecel Planning, Case Studies in Mexico. North-Holland,
Amsterdam. 10. Kornai J and Liptak TH (1965) Two level planning. Econometrica 33( [ ), 141-169. l l. Kornai J (1969) Man-machine planning. Econ. Phmn. 9(3) 204-234. 12. Land AH and Powell S (1973) Fortran Codes fi~r Mathematical Progratnming: Linear, Qua&'atic and Discrete. Wile,,', New York. 13. Lasdon LS (1970) Optimi.zation Theor.v j'or Large Systems. Macmillan, New York. 14. Obel B (1981) Issues of Organisational Design. Chaps
l-3. Pergamon Press, Oxford. 15. Obel B (1981) Issues of Organisational Design. pp. 197-200. Pergamon Press, Oxford. 16. Obel B (1981) Issues of Organisational Design. pp. 195-197. Pergamon Press, Oxford. 17. Provopoulos G (1982) Public enterprises and organisations. Economic Theory and Greek Practice. Institute of Economic and Industrial Research, Athens. 18. Samouilidis JE and Arabatzi-Ladia A (1982) A Model for the Greek energy system. Eur. J. Opl Res. 9, 144-160. 19. Samouilidis JE (1982) A systemic approach to energy policy. In Proceedings of the First National Congress for Soft Energy Resources. Institute of Solar Technology, Thessaloniki, Greece, October 20-22, pp. 29-45. (In Greek). 20. Samouilidis JE (1982) Systems analysis for policy formulation. Paper presented at the Joint Congress of the Hellenic Operational Research Society attd the Greek Management Association, Delphi, Greece. (In Greek).
21. Samouilidis JE, Berahas SA and Psarras JE (1983) Centralized versus decentralized decision making. Energy Poli
ADDRESS [--OR CORRESPONDENCE:
APPENDIX
I
The decomposition principle
1. Atkins D (1974) Managerial decentralization and decomposition in mathematical programming. Opl Res. Q. 25, 615-624. 2. Burton RM and Obel B (1977)The multilevel approach to organisational issues of the firm--A critical review. Omega 5(4), 395-414. 3. Burton RM and Obel B (1980) The efficiency of the price, budget and mixed approaches under varying a priori information levels for decentralized planning. Mgmt Sci. 4, 401-417.
Application of the linear programming method to larger and more complex problems, in connection with the limited capabilities of computers in the early sixties, introduced the use of the DP in the development of special algorithms, adapted to the characteristics of each particular type of problem [4]. It was soon recognized that the DP apart from helping to overcome computational difficulties,could also be employed in the development of models for decentralized planning in multilevel organizational systems.
Samouilidis, .4rubutzi-Ladia--Decentrali-ed
446
The t'ollov, ing example shov, s a simplified representation of the problem in a two-level system, frequently met in the literature. max CI .V~ + C: .V:... Cp,Vp
Decision Making in the Energy Sector subject
to
alex I + a~: X, + aI3 fl + at-t Y: + a:5 Y; <2 C
I
a:t ,~(i + az2 ,'(2 <-- C,
(1)
ajt.,~'~-'-a3.-X 2 N C 3
subject to At .t'l + A,.X: . . . . 4pXe = <_bo B~ .V~ = < b I
B , X , = <_b2
B.,V~ =
a~, YI + a u I"., + a45 Y3 < C~
(2~
(3)
<_be
A7 > 0 . i = 1 . . . . p where:
Ci, A'i, bi are vectors Ai. Bi are matrices It can be seen that the problem has a block-angular structure appropriate for decomposition and can be divided into ( p + l) subproblems. Each one of p subproblems corresponds to the activities f i of subsystem i. and the remaining subproblem, called the master problem, corresponds to the Central Unit of the system and co-ordinates the solutions of the p subproblems so that coupling constraints (2) are satisfied. The main difficulty in solving this type of problem is the co-ordination between the master problem and the subproblems. Algorithmically, this means the sending up and down of values of selected parameters between the master problem and the subproblems. Organizationally co-cordination means a two-way transfer of information and c o m m a n d s . There are two main approaches to the problem. The pricing approach whereby the information passed by the master problem to the subproblems concerns prices--the Dantzig-Wolfe algorithm [5] i,; the best-known representative in this area. The budgeting approach whereby the information sent down concerns resource allocations--the Kornai-Liptak algorithm [101 is representative of this area. Comprehensive reviews of the literature on the subject are given in [2, 3]. The objective of the problem is to find a feasible and, if possible, o p t i m u m solution. An important criterion in evaluating an algorithm is the rate of convergence during the first interactions, and its importance is enhanced in applications concerning decentralized planning in multilevel organizations. Research results have shown that the rate of convergence depends on the organizational structure of the system and in particular on:
where the P ' s are profit coefficients, a,j's are input/output coefficients, C's are fixed resources, and X's represent decision variables for subunit 1 and Y's for subunit 2. It is the special structure of this problem which lends itself to the decomposition and the analogous hierarchical model of the sector. First. note that the objective function (for profit) is linear, and therefore separable, and thus the portion with the X's can be assigned to subunit I and the Y's to subunit 2 without interference (or externalities). Next, the first constraint affects all decision variables of the sector and is called the c o m m o n constraint. The next two constraints involve only the X's and the last constraint only the Y's. These are called subunit constraints. Similarly. C~ is called a c o m m o n resource, and C,, C 3. Ca subunit resources. The organizational algorithm consists of the executive, or co-ordination unit, solving a problem and then passing information to the subunits. The subunits, in turn, solve a planning problem and send information back to the executive. For a linear programming model, each submodel is also a linear program for the Dantzig-Wolfe approach terminating at an overall optimal solution after a finite number of iterations. This process is given in more detail: let us begin with any subunit problem at an arbitrary iteration, e.g. subunit 1 is asked to solve the following LP: max R~ .V~ + R: ,g., with respect to
A"I, X~.
a21X ~+ a2,,.X: <, C, a3zX I -- a32X 2 <_.C 3 where R~ and R 2 are modified profit co-efficients which are communicated to subunit I by the executive or master program. Subunit I then, sends the optimal values of ,Y~, X2 back to t h e executive unit. The executive program consists of the original objective function and the c o m m o n constraints plus convex combination constraints. The program determines the best convex combination of all past solutions sent up by the subunits. The co-ordination program is: max
- - T h e n u m b e r of coupling contraints. - - T h e n u m b e r of subproblems. - - T h e initial solution.
(2)
subject to
f ~,(P~ X~ + P: X') + v,(I)3Y{ + P, Y',_+ P~ Y~ ) ¢=I
with respect to ]i T,
~ t 1, . . . .
t' I ....
, t"T
subject to APPENDIX
T
II
7. ~z,(at. X'~ + at,X',_) + c,(a,3 g'~ + at~ Y', + a~5 g~) < C~ (3)
The Dantzig-Wolfe algorithm Consider that the energy sector has a two-level structure, i.e. a central unit (Ministry of Energy) and two peripheral subunits, subunit 1, e.g. Public Power C o m p a n y and subunit 2, e.g. State Oil Refineries. The mathematical model which represents the decision problem is given as follows: max P = Pt X, + Pz'V: + P~ Yt + P~ Y'- + P~ Y~
1"
~ ,u,= 1 t=l r t=l
#~, t'~ > 0 Vt
with respect to
Yl, X:. YI, Y'., r'3
t=l
(1)
The decision variables are the linear weightings lL's and o's. The solutions to the organizational decision problem ,g"
Orneea, Vol. IZ No. 5 and YJ are deri',ed from the master problem. The? are v*ritten fully as: .Y~ = .a,..F,I - . . . + #,.'(,r and I"~-= t', F) - : - . . . ,
t': F,r
where the superscript 7" refers to the optimal solution. The master primal p r o g r a m is not of great interest. The interesting information is in its dual. The dual variabte for the c o m m o n constraint, call it =t for C~, measures the marginal value, an o p p o r t u n i t y cost of that c o m m o n resource. Subunits which use this resource are then "charged' for its use by substracting the "cost" of the resource from their profit coefficient. The modified profit
447
coefficient for )(k is p v e n as: R~ = R I - - a , l ~ l . T h a t is, the net profit coefficient for ,~'~ is the original coefficient minus the cost of the use of the scarce c o m m o n resource. Similarly, modified profit coefficients are calculated for X, and Y's. Briefly, the executive sends modified prices (hence the term pricing approach) to the subunits, the subunits calculate their best solution for outputs, and send these back to the executive who in turn determines the value of the scarce corporate resources, and adjusts the prices. The executive provides a solution for prices, the subunits t'or outputs.