Goal programming & data envelopment analysis (GoDEA) for target-based multi-level planning: Allocating central grants to the Greek local authorities

ELSEVIER European Journal of Operational Research 87 (1995) 535-550 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Goal programming & data envelopment analysis (GoDEA) for target-based multi-level planning: Allocating central grants to the Greek local authorities Antreas D. A thanassopou los Warwick Business School, University of Warwick, Coventry CV 4 7AL, UK Received November 1994; revised May 1995 Abstract In this paper we develop an interface between Goal programming and Data Envelopment Analysis (GoDEA) in order" to integrate target setting and resource allocation in multi-level planning problems. Data Envelopment Analysis (DEA) has been traditionally used for assessing the performance of individual decision making units and, therefore, necessary extensions are needed to apply DEA principles to the global organisational level without, however, losing its attractive features. The method was originally developed as an aid to the reorganisation of the allocation of central funds to local authorities in Greece. Keywords: Multi-level planning; Goal programming; Data envelopment analysis; Local authorities; Equity; Effec- tiveness 1. Introduction: Motivation Production systems which are divided into sub- systems (for example, regions within a country, countries within the European Community, re- gions within a firm and branches within a region of the firm) constitute multi-unit, multi-level planning systems. The complicated structure of organisations operating with multiple levels of de- cision making requires effective planning and co- ordinating mechanisms to resolve the conflicting objectives/interests of interested parties within the organisation. In this paper a revisit to multi-level planning 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights SSDI 0377-2217(95)00228-6 problems is attempted from an applied perspec- tive. The attempt was motivated by an ongoing project for developing a budgeting system for Local Authorities in Greece. The planning frame- work put forward in the paper, namely goal pro- gramming & data envelopment analysis, is seen used as an instrument of Group Decision Making. The framework seeks to combine conflicting ob- jectives of resource allocation, namely efficiency, effectiveness and equity, and also to encapsulate perspectives of different levels of management in the generation of planning scenarios. The rest of the paper is organised as follows. A review of existing approaches of multi-level reserved 536 A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 planning is provided investigating how effective these methods are in terms of incorporating plan- ning objectives. The rationale of target-based multi-level planning models is discussed leading to its mathematical representation which com- bines features of goal programming and data en- velopment analysis (GoDEA). The GoDEA model is used then to facilitate the development of a decision support system of financial planning in local authorities in Greece. 2. A review of multi-level planning systems The case of planning in multi-unit multi-level organisations (MULOs) is an old one with an extensive literature reviewed inter alia by Nij- kamp and Rietveld (1981), Sweeney et al. (1978) and Nachane (1984). Ruefli (1974), in his review, raises issues concerning the formulation process of multi-level planning problems and the limi- tations of the existing methods to address real life problems (an example is Goreux and Manne, 1973). These studies are based on the assumption that the design of large planning systems can be made by decomposing them into a number of smaller subsystems each with its own goals and policies. As Anandalingam (1988) argues the problem in a multi-level system is how to ensure that all decision makers, acting according to their own goals, will achieve overall system goals. Nijkamp and Rietveld (1981) describe three principal prob- lems of policy making in multi-level environ- ments: • Interdependencies between the components of the system; • Conflicts between various priorities, objec- tives and targets within individual components of the system; • Conflicts between the priorities, objectives and targets between the various components of the system. Based on these three characteristics, Nijkamp and Rietveld (1981) advocated the usefulness of multiobjective programming methods for ad- dressing planning problems in MULOs. The achievements of conflicting objectives, at the glo- bal level, can be compromised using multiobjec- tive programming methods whilst conflicts be- tween the organisational levels of the system require coordinating mechanisms so that all levels act in agreement. The exact form and character- istics of these coordinating mechanisms is an issue open to different interpretations. Two alternative schools of thought have emerged in the literature for solving planning problems of MULOs. Ruefli (1974) calls them the classical and behavioural models, Sweeney et al. (1978) call them decompo- sition and composition approaches respectively, and Burton and Obel (1977) call them the pricing and the budgeting approach to multi-level coordi- nation. Burton and Obel (1977) provide a critical re- view of the two approaches examining their appli- cability and organisational implications. They argue that both approaches have seen sporadic applications in agricultural and macroeconomic planning on an ad hoc basis. Simulation results are more commonly used to test the behaviour of alternative coordinating mechanisms. On the organisational side, Burton and Obel (1977) re- cognise the gap between the theory of multilevel planning systems and organisational practice. 3. Target based multi-level planning As mentioned earlier, multi-level organis- ations are organised into independent subsystems based on geographical and/or functional seg- ments. For example, the education system in a country has geographical subsystems, e.g. local educational authorities in the UK, and also func- tional subsystems, e.g. primary and secondary education. The structure of multi-level multi-unit organisations affects considerably the quantita- tive representation of planning objectives. For instance, increased status of authority to lower levels of decision making may lead into time de- lays and internal conflicts with higher levels of management. On the other hand, highly central- ised systems may find it difficult to motivate lower level management to participate in the achieve- ment of organisational plans formed without their consensus . A.D. Athanassopoulos I European Journal of Operational Research 87 (1995) 535-550 537 3.1. Objectives of multi-level planning The question of planning in MULOs concerns the best deployment of all types of resources in order to meet the organisational objectives. There is a technical and a philosophical side in selecting resource allocation criteria. The former often relates to information and data availability whilst the latter relates to objectives and purpose. Equity, efficiency and effectiveness are generally said to reflect the organisational mission in non- profit, multi-unit organisations. This representa- tion would also guide decision making towards: • Maximising the global achievements of the system (effectiveness); • Maximising the contribution of individual units to global targets (efficiency); • Maximising the share of each individual unit to the allocated resources (equity). In a seminal article, Savas (1978) sought to open a debate concerning the objectives in the provision of public services. Savas (1978) dis- cusses the importance of equity as a resource allocation criterion whilst urging for deeper man- agement science involvement to improve the re- presentation of this objective in the allocation of resources. Undoubtedly, the representation and measurement of equity in allocating resources is still an open issue with technical and political dimensions (Mandell, 1991). Apart from the representation of these objec- tives, a more important challenge is on how to monitor their simultaneous satisfaction in the planning process. In practice, there are signs of overweighing some of these objectives at the ex- pense of others. For example, the national health system in the UK has shifted progressively from an equity based to an efficiency based system of resource allocation since the development of in- ternal and contract markets. Equity as sole crite- rion does not guarantee the best possible use of resources whilst efficiency in itself can increase inequality in the provision of public services. Centralised resource management is con- sidered as the process where central management is responsible for the allocation and control of resources allocated to individual decision making units (DMUs). Since these resources are obtained Global Global inputs outputs ~k Central '~ organisation DM 1 :DMUj '~ o,= DMU • =- '~ uts Outputs uts Fig, 1. Rationale of the centralised planning system. Maximtse GLOBAL targets' achievement Maxlmise DMUs' contribution to GLOBAL targets by central means (e.g. taxation) there is a need for public accountability regarding their utilis- ation. Central co-ordination of the allocation pro- cess, but more importantly auditing of the actual use of these resources by individual units, is the typical route followed by central governments. On the modelling side Thanassoulis and Dyson (1992) provide a sufficient basis for estimating efficient input/output targets for individuals. Their formulation is not sufficient, however, to address planning and resource allocation prob- lems where all DMUs of the organisation need to be considered simultaneously. This enhancement would give the opportunity to accommodate glo- bal organisational targets, global resource con- straints and finally to reinforce internal communi- cation between DMUs (resource re-allocation). A pictorial representation of this planning system is given in Fig. 1 and will be used as the basis for discussing the rationale of the centralised target- based planning system. The MULO represented in Fig. 1 consists of a central coordinating mechanism that is responsi- ble for controlling/allocating global resources to DMUs operating similar, but independent func- tions. Central management seeks to maximise the achievement of global input/output targets (as- suming for the moment they are known). Individ- ual DMUs are expected, therefore, to maximise their contribution to the achievement of global organisational targets. This conceptual frame- work used to describe the operations of MULOs leads inevitably to questions concerning the oper- ationality of the system, the assessment of global and DMU based targets and finally the manage- 538 A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 ment of interactions (resource re-allocation) be- tween DMUs. The development of this planning framework will embark from the assessment of performance targets for individual DMUs proposed originally by Thanassoulis and Dyson (1992). The targets of DMU k that uses quantities of inputs x k, i = 1 . . . . . m, to deliver quantities of outputs y k,, r = 1 . . . . . s, are given by the linear programming model in (1). k k erZr-- E PfOf Vk Max r~Oc i~lc subject to =zryr , rEO~, Vk, j=l ~k xi j k k =O~x~, i~ Ic , Vk, ]=1 6~yrj>-- y~, rE Of, Vk, j= l ~kXi j~Xf , i~ I f , Vk, j=l where: Xu, yrj P~, P~ (1) is the level of input i and output r of DMU j respectively. is the factor contribution of DMU j to the targets of DMU k. are user specified preferences over the improvement of inputs/outputs of DMU k. I-= Ic U Of is an index set representing inputs 1=1 . . . . ,m 0 - Oc U Of is an index set representing outputs O=l , . . . , s , Ic, Oc are subsets of inputs/outputs to be im- proved, If, Of are subsets of inputs/outputs not prior- itised to improve. The solution of (1) would yield targets for the inputs/outputs of each DMU in isolation. This, however, does not provide sufficient insights into the achievement of global organisational targets. A simultaneous representation of all DMUs within the planning process of the MULO, as advocated in Fig. 1, is necessary and this has been achieved in the first set of constraints (2a) of the planning model that is listed in (2). The activities of a multi-unit, multi-level or- ganisation can be aggregated and displayed by global levels of inputs/outputs, which are allo- cated among or produced by individual operating units. The extent to which the organisation achieves these global targets is considered as a surrogate measure of its operational effectiveness which can be supported by the efficient contribu- tion of individual DMUs (see (1)). The oper- ational effectiveness is represented in the plan- ning model with the set of constraints (2b). The planning process within a MULO is also subject to policy making constraints. As an exam- ple, a set of balance of payment constraints are proposed (in (2c)) for linking the aggregate target achievements of commensurate inputs and out- puts. These constraints can be used, for instance, to balance the income-expenses relationship in macroeconomic planning models of local govern- ment spending. The earlier discussion on the rationale of the centralised target-based planning model will lead now to its mathematical representation in (2). Some additional notation to the one used in (1) is necessary to facilitate the mathematical formu- lation of the centralised target-based planning model. • At the global organisational level it is antici- pated that for a subset of controllable inputs lv and outputs Ov management will be able to spec- ify desired global targets whilst the global levels of the remaining inputs iv and outputs Ov will be estimated by the solution process of the model. It is expected that the distinction of the global controllability of inputs and outputs will apply to inputs and outputs classified as controllable (lc, Oc) at the individual unit level. Thus the dis- tinction of inputs/outputs can be stated as fol- lows: LuL~L and 0,, O Ov =-- Oc. A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 539 • Finally, the use of balance constraints that will be used to link the estimated global targets for commensurate inputs and outputs will apply to a subset of such inputs (IB c 1) and outputs (08 c 0). Using this notation the centralised target-based planning model is formulated in (2) as a goal programming-based problem. Min Pi,Pr, ni, nr, di, &, ,Sj Goal programming & Data Envelopment Analysis planning model {j~ ~Ic (pn nJ ~_p F PJ/t 1 i Xij X i j / pp~ i = i r e Oc Yrj Vrj/ ~ I "g 'd ie~- E "gr d r } v 'ox , , ov (2) subject to Representation of individual DMUs: ~f y,j - pkr + n, = y ~ , r E Oc, Vk, j=l - i 6fxii+P~--nki =--xki, i e l c , Vk, j= l j '=l -- i 6f Xij >-- -- X k, i E If, Vk, (2a) j= l Effectiveness and global targets'achievement: - ~ 6]xq . . . . . ~ 6,xq+d +=-GX~ j=l j=1 ViEL , - . . . . . + vx ,=0 j=l j=l ViE iv, 6]Y~i+" '+ ~ 6;Ya +d~-=GYr j=1 j= l Vr E Or, ~ a}yr j+ ' ' '+ ~ 6~y, j -VY ,=O j= l j=t Vr E Ov, (2b) Budget balance: E E +. . . + 87)x,, i~ lB j - - 1 - ~, ~ (a l+ . . .+6~)y , . j O, , " i, "" r~ t" i, 1-" r~ VX~>-O ViE[~, VY~--- 0 Vr E 0~, (2c) where: ni,p'; are negative and positive deviation variables for the input i of DMU j, n~,p~ are negative and positive deviation variables for output r of DMU j. d~- ,d/ are the positive and negative devi- ation variables from the global target of input i and output r. P~, PP preferences over the minimisation of positive/negative goal deviations of input i. P~, PP preferences over the minimisation of positive/negative goal deviations of output r. Pg, Pg are the preference levels related to the global target of input i and output r. x u, Yrj is the input i and output r of DMU j. GXi, GYr are the i-th input and r-th output glo- bal targetlevels with prior knowledge. VX~, VYr are the i-th input and r-th output glo- bal targets without prior knoledge of their values. B is a user specified constant concerning the balance between commensurable inputs and outputs in the planning model. IB, OB are the subsets of commensurable in- puts and outputs. The model in (2) is a goal programming one. Goal programming has been used previously for assessing performance of individual DMUs by Charnes et al. (1988) and Thanassoulis and Dyson (1992). However, these models aimed at assessing performance of individual units assuming some 540 A .D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 knowledge of 'ideal' input/output targets. The formulation in (2) provides a framework with no predetermined assumptions on individual units' achievements and has a structure that is described in more detail next. 3.1.1. DMU representation in the GoDEA model Each DMU in (2) is represented by its own input/output constraints. For example, the con- straint set (2a) represents DMU k. This constraint set represents the comparisons made between the inputs/outputs of the assessed DMU k, k k (xi,yr), and its composite unit (Ej6~x~ i, Zj 6~yrj). The formulation in (2a) differs from the target setting model in (1) insofar as the pres- ence of the goal deviation variables for inputs n/k,p~ and outputs k k n~,pr of DMU k are con- cerned. The allowance given to over and under achieving input/output 'goals' in (2a) has reper- cussions on the estimated targets of individual DMUs. As is discussed later, suitable formula- tions of the objective function of (2) can yield input augmentation and/or output reduction tar- gets in the light of supporting the achievement of the global organisational targets. The representation of individual DMUs using separate sets of constraints in (2a) would also yield (in the optimal solution) efficient facets for projecting inefficient DMUs. The efficient facet of DMU k for example will be identified from efficient DMUs j with 6~* > 0. The selection of efficient facets, however, is driven by the nature of the objective function in (2) which seeks to facilitate the achievement of global targets. This is in contrast with ordinary DEA (see (1)) where a DMU is projected on the facet of the efficient frontier that would minimise the distance be- tween its current performance and the one its should have had it been efficient. This implies that inefficient DMUs may be projected on differ- ent efficient facets by the solutions of models (1) and (2) respectively. 3.1.2. Estimating global input~output targets A significant component of GoDEA is the in- corporation of global target levels for controllable inputs and outputs. The nature of these global targets, however, needs further elaboration. Pre- vious studies on multi-level planning models, al- beit recognising the importance of the issue, as- sume prior knowledge of global target levels, Freeland and Baker (1975). In practice, however, organisations do not have well-defined targets for all inputs/outputs simultaneously and also it is difficult for management to predict intuitively the impacts of various planning policies over organi- sational outcomes. The global input/output targets are repre- sented in (2) using the constraints in (2b). These constraints seek to aggregate the contribution of the composite unit of assessed DMUs, say DMU 1, for the i-th controllable input (Z~'= 1 ~)xij) and r-th controllable output (E~'= ~ 6)yr/). A further distinction is taken into account by declaring in- puts/outputs (Iv, Ov) with prior knowledge of their global target levels and those, (/v, Ov), that will be treated as variables in (2). Further elabor- ation needs to be made concerning the possible ways of estimating global input/output targets (GXi, GY,.) prior to the solution of (2). For those inputs/outputs where the estimation of global targets is desired prior to solving GoDEA, a more systematic process can be em- ployed based on the concept of structural effi- ciency introduced by Farrell (1957). Farrell ar- gued that efficiency targets within industries at the corporate level can be obtained using those firms (DMUs) with the best observed practice. This type of analysis was called structural effi- ciency and it has been extended by Forsund (1992), Giokas (1991) and Athanassopoulos and Ballantine (1995). Systematic estimates of global performance targets for inputs (/~) and outputs (Or) are given in (3). t l GYr = ~r j= ! t l GXi=0*( ~ xij) Vi~lv, \ i = 1 (3) where the targets are obtained solving the se- quence of linear problems in (4)-(4') (one for each controllable input/output). A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 541 DEA for global (structural) performance targets Max 2 (4) "/j, ~r subject to j= l j= l j=! n j l n YJYd = Zr E yrj, j= l YjYri >- Z, yrj, j= l rEOv, r4=f rE OfU Ov j= l Max - t},. yj, Oi subject to ~/jXij ~ -- ,~ Xi j , j= l i E I . (4') TjXTj = Xf j =1 j 1 TjXi j = Oi ~ Xi j , i E lv, i 4: t =1 j= l ~, yjyrj>--~ yrj, rEO =1 j= I -- i ~/jXij ~ -- i Xij , i E lf U [v. j= l j= l The linear programming problems in (4)-(4') seek to assess global performance targets for each controllable input/output. The estimation of these targets requires solution of as many linear problems as the number of controllable inputs Iv and outputs Ov. The targets in (3) can be used to provide quan- titative estimates of global targets for selected inputs/outputs used in the GoDEA model in (2). As the global targets of each input/output are obtained from the solution of different linear pro- gramming problems it will not be feasible, in prin- ciple, to achieve these extreme targets simulta- neously. The solution process in (2) will propose compromise sets of global targets in the light of decision makers' preferences. 3.1.3. The objective function of GoDEA The objective function of GoDEA contains the deviation variables used in the constraint sets (2a), (2b) and (2c), and seeks to minimise devia- tions that correspond to the global and individual DMU targets. All deviation variables, included in the objective function, have been standardised in a per unit of input/output basis. The first part of the objective function includes the deviation variables from global input/output targets, Min ~ p~ + d[ + ~ P7 d___~_~. di, dr i~ 1,. GXi r E o, GY," The priorities used in the objective function re- flect the penalty per unit of deviation from the global targets. It is noteworthy, however, that the deviation variables used for global inputs (out- puts) represent an overachievement (under- achievement) of the original input (output) global targets. An assumption is made, therefore, that individual DMUs would seek in principle to use more resources than available and to deliver less services than are desired by central management. This assumption can be relaxed by simple modi- fication of the goal deviation variables. The second part of the objective function in- cludes the deviation variables of inputs and out- puts of individual DMUs, n " i Pi, Pr, hi, nr j : 1 i c Xij Xi / j = l , ~ yrj yr / The priorities attached to these deviation vari- ables can be used to monitor the contribution of individual DMUs to global organisational targets. Under- and overachievement goal deviation vari- ables, n{,r,pJ,r are used for controllable inputs/ outputs of each DMU. The presence of two-way deviation variables implies that the problem can be solved by under- or overachieving the ob- served input/output values of individual units. This is a fundamental departure from DEA models which assume that the assessed targets should always contract inputs and expand outputs. 542 A .D. Athanassopoulos / European Journal of Table 1 Developing planning scenarios Preferences in GoDEA Planning strategy (model (2)) (i) Output expansion & input contraction (ii) Output & input expansion (iii) Output & input contraction P~, P~P'< 0 and P[, P~->O P,P, Pr p --< 0 and pn, p~ ::.. 0 P~,P~ A.D. Athanassopoulos / European Journal of 543 soclodemographlc variables • - " " " " Bui l t -up area Heavy Industr ial ~ - size Actual Genera l area Total size of • , households Fiscal capacity house area. • • / costsLab°ur ~ ~ Central Grants Serv ice , ~ Fees & Charges expeo~s---- LAs • " . . . ' ~ Investments Ma intenance ~ " " costs ~ ~ Rate of charges Loans . . . . . F lnag¢ la l Eny l ronment . . . . . . Fig. 2. Input/output framework of local authorities (LAs) operations. nancial performance of local authorities rather than on the provision of specific services (e.g. sanitation, road lighting) provided. An input/out- put model was developed to represent demand and supply of services using surrogate measures of demand (e.g. population characteristics) and financial measures of supply. Fig. 2 provides a pictorial representation of the financial planning framework used in the analysis. The framework described in Fig. 2 contains sociodemographic and financial variables of local authorities. These variables will be used to de- scribe the operating profile of local authorities and also to represent quantitatively the objectives of resource management, namely effectiveness, equity and efficiency. Socio-demographic variables are used to cap- ture the needs of local authorities to provide ser- vices and, thereby, to reinforce the equity prin- ciples in the allocation of central grants. The implied assumption within the model is that local authorities with similar demand for services should also have similar financial performance (e.g. central grants). As the demand factors are beyond the control of individual local authorities no immediate improvement in their level will be sought by GoDEA. Had the project been given reorganisation scope the latter assumption could have been relaxed. Summary information of the socio-demo- Operational Research 87 (1995) 535-550 Table 2 Socio-demographic demand factors of financial planning Variable Median Max Min No. of actual households 16 931 377 929 9 210 Built-up area (km 2) 5 698 38 300 918 General area (forests, 3 640 133 600 0 agriculture, etc.) (km 2) Heavy industrial use 68 609 2 194 356 13 290 area (m 2) Estimated value of 88 157 56 property (103 Drch/m 3) Total size of house 1 387 007 2 586 433 674 735 area (m 2) graphic variables of local authorities is given in Table 2. The number of actual households represents an estimate of the true population of each local authority and is based on these households with a positive consumption of electricity energy. Built-up area seeks to represent the effect of size on the provision of services, whilst the industrial use area seeks to complement the provision of services concerning environmental issues. An- other surrogate measurc of demand is the total size of houses of each local authority. This vari- able, in conjunction with the built-up area, gives an indication of the population density, and is also used as the basis for estimating the potential for collecting local fees and charges. An important element of the planning model is the assessment of the fiscal capacity of citizens of individual local authorities to be charged for local services. This was done using an estimated value of the properties within each local auth- ority. Each local authority is divided into geo- graphical segments and a different value is at- tached to each segment by the national tax office. These differences can be substantial and, there- fore, some type of distribution of different pro- perty values was used to give a fair representation of the fiscal capacity. This was done by weighting the coefficients of the value of property with the proportion of the area within the local authority that this coefficient was applied. The financial factors in Fig. 2 display types of expenses and sources of revenue of local authori- ties. These variables need to be used in an input- 544 A .D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 output framework within the formulation of GoDEA. The economic objective of each vari- able, notably maximisation or minimisation, will depend on the level of management that is re- sponsible for the planning process. For example, the central government seeks to minimise the level of central grants allocated to local authori- ties whilst the opposite objective is sought by local authorities. On the other hand, the government encourage local authorities to maximise the level of fees and charges collected whilst individual local authorities may be reluctant to support this policy. The solution process .discussed next ex- plores the implications of adopting local or cen- tral government scenarios in solving the GoDEA model (2). 5. Solution process of GoDEA The implementation of the GoDEA model consists of a number of interrelated stages that need to be completed prior to solving the prob- lem. These include: • selection of a panel of decision makers from the central and local management; • assessment of global target values for con- trollable inputs/outputs; • allocation of resources and assessment of targets for each local authority; • generation of alternative scenarios using sensitivity analysis. Two panels of decision makers were selected to facilitate the solution process of GoDEA. The first is made up of five councilors of the associ- ation of Greek local authorities (elected body), whilst the second is made up of three advisors of the minister responsible for local authorities. The application included the largest sixty-two local authorities in Greece (population over 20 000) which have a satisfactory degree of compatibility in size and range of activities. 5.1. Financial planning scenarios and decision makers' preferences The development and solution process of GoDEA requires preferences from decision mak- ers over the achievement of different planning goals. The current application requires represen- tation of two main groups of decision makers, namely central and local government. The devel- opment of systems that quantify decision makers' preferences is an involved problem. Experimental studies by Schoemaker and Waid (1982), Buch- anan and Daellenbach (1987) and Khorramshah- gol and Moustakis (1988) have investigated the merits of techniques such as the Analytical Hier- archy Process, The pairwise comparison, and Re- gression Analysis without, however, conclusive evidence on the superiority of one over the other. A regression based methodology, namely least absolute deviations (LAV), was selected for ob- taining preferences to be used in the objective function of GoDEA. The LAV method has been developed by Charnes et al. (1989) and Sueyoshi (1994) and its mathematical formulation can be found in the Appendix. The solution of the LAV model was im- plemented as follows. A set of ten local authori- ties with their standardised scores on inputs/out- puts was selected (without revealing their ident- ity). Each decision maker was asked to rate how successful was the financial profile of each LA by assigning a value from 0 to 100. These values follow a monotonically increasing order (the higher the better). The subjective ratings assigned by decision makers were used as dependent variables of the LAV regression model, whilst the standardised scores on controllable inputs/outputs were used as independent variables. Two separate LAV models were employed, one for the governmental preferences and one for the local authorities re- spectively with the estimated weights reported in Table 3. The results listed in Table 3 have been sub- jected to sensitivity analysis tests seeking to exam- ine how flexible the assessed weights are on fluc- tuations of the DMs ratings. The weights assessed on individual financial variables will be used as priorities in the objective function of the GoDEA model. The magnitudes of these weights reflect the relative importance of the corresponding cri- teria, whilst their sign indicates whether the under or over achievement goal deviation variables should be included in the objective function. A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 545 Table 3 Planning scenarios and preferences over financial variables Governmental Local authorities' Expenses ~ eights ~ weights u Labour costs - 19 -9 Maintenance & stocks -11 +3 Service expenses + 16 + 18 Investments +8 + 14 Income Governmental grants - 12 +26 Fees and charges +25 +8 Loans + 14 + 10 Rate of charges N/A + 12 E ] weights I 1(10 100 a Preferences expressed by three governmental officials. b Preferences expressed by five elected councilors. Negative weight, in Table 3, for income finan- cial variables indicates that the objective function of GoDEA should maximise the variable of nega- tive deviations in the objective function of GoDEA. On the contrary, a positive weight for income financial variables indicates that the ob- jective function should maximise the positive devi- ations in the objective function. An inverse re- lationship holds for financial variables representing local authorities' expenses. The two sets of preferences are applied to the goal vari- ables of individual LAs or the global targets de- pending on whether the governmental or LAs scenario has been considered. 5.2. Estimating global targets The model in (4) was employed to estimate global input/output targets of local authorities. All inputs/outputs listed in Fig. 2 were included in the model apart from the rate of charges of local fees and the average value of property within each LA. These variables were discarded from model (4) as it yields aggregate input/output targets for LAs and therefore it cannot accommo- date variables measured in a ratio or percentage form. Model (4) was applied eight times giving pre- emptive priority to the optimisation of each con- trollable input/output factor at the aggregate or- ganisational level. The targets of governmental spending were estimated twice, once to find their maximum feasible level (local authorities' sce- nario) and then to find their minimum feasible level (central government scenario). Table 4 lists a payoff table of the rate of, optimal pre-emptive, expansion/contraction of inputs/outputs. The bold entries of Table 4 show the coef- ficients of maximum feasible contraction (0") or expansion (g*) of controllable inputs/outputs. For example, when governmental grants are sought to be maximised, an increase by a factor of 2.51 is found feasible by the model in (4). On the other hand, the minimisation target for the same objective gave a 0.66 rate of reduction whilst targets were also obtained for maintenance Table 4 Payoff table of global targets using pre-emptive priorities over inputs/outputs Implications on other objectives Objective Govern. Service Fees & optimised grants Wages Loans Maintenance expenses charges Investments Gov. grants a 2.51 0.89 0.43 3.36 1.41 1.03 1.53 Gov. grants b 0.66 0.84 1 0.78 1 1 Wages 1 0.6 0.857 0.87 1 0.8 0.81 Loans 1 0.56 0.04 0.99 1 1 0.31 Maintenance 0.45 1 1 1 0.079 0.43 0.94 Service expenses 1.09 1 1 0.4 1.47 1 1 Fees & charges 1 1 i 1.08 1 1.84 1.22 Investments 1 1 1.62 1 1 1.09 2.13 1 a Maximise central grants (local authorities). b Minimise central grants (central government). 546 A .D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 (contraction by 0.87) and service expenses (con- traction by 0.78). Worth noting are the economic implications conveyed by the assessed targets for individual inputs/outputs. For example, in the last row, when the maximisation of investments is targeted, the increase of investment spending yields an expansion target of the loans obtained by the LAs by a rate of 1.62. The results from the overall targets in Table 4 indicate the diversity in the assessment of global input/output targets as a function of differential pre-emptive preferences. Simultaneous achieve- ment of the targets listed in the diagonal elements is not feasible and, thus, compromise strategies need to be explored using the GoDEA model for maximising the achievement of these targets in the light of decision makers' preferences. Results obtained from the solution to the GoDEA model are discussed next by concentrat- ing on three main areas; global targets achieve- ment, reallocation of inputs/outputs among indi- vidual local authorities and sensitivity analysis of the planning scenarios. 5.3. Target setting at the global local authority level GoDEA was used as a tool of generating alter- native planning scenarios to facilitate negotiations between local authorities' and central govern- ment representatives. Two solutions were ob- tained representing central and local govern- mental views insofar as the levels of global inputs/outputs and also the preferences (see Table 3) over input/output improvements were concerned. The part of GoDEA in (2b) included three controllable inputs (wage expenses, maintenance and loans) and three controllable outputs (local fees and charges, expenses on investments and provision of services). Governmental grants were used as an input to be minimised in the central governmental scenario, and as an output to be maximised in the local governments scenario. The desired global target levels were obtained in all cases from the targets listed in Table 4. Finally, the constraints in (2c) sought to balance the ex- penses with the income sources of local authori- ties at the aggregate level. Results on the achievement of aggregate tar- gets assessed for local authorities are given in Fig. 3 representing the policies of central government and local authorities respectively. The solution of the two models was made using the AIMMS modelling system, Bisschop and En- triken (1993). The two scenarios gave diverse re- sults insofar as the targets' achievements are con- cerned. This was due to the different weights used in the objective function, as well as, the different level of global targets for central grants between local authorities and central government. Each input/output variable is represented in Fig. 3 with three target values. For example, the wages variable has current levels of 2 026 510 500 drachmas which would increase by 30% according to the LAs scenario (column two) and decrease by 8% according to the governmental scenario (column three). These values, however, should not be looked in isolation as they depend on the varying aggregate levels of income, as well as the varying expenditure mix of each scenario. Governmental spending is a controversial vari- able as its overall level affects the activity levels of all other variables. The achievement of the very high goal assessed by the maximisation of governmental spending leads to increased levels of spending on wages, services and investments. At the same time the local authorities' scenario yields maximum achievement of the target level for fees and charges. In summary, this is a growth 10o6 Drachmas 50 40 30 20 10 Wages Maintenance Grants Loans Services Investments Fees [==Observed IlL. Authorities BIGovamment] Fig. 3. Aggregate planning scenarios for the local authorities. A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 547 scenario which could be implemented under cer- tain general economic conditions that would en- able the government to increase spending at such a high level. It could also be connected with an expansion of the activities of local authorities in new areas implementing a decentralisation policy. The governmental scenario is, as expected, more conservative regarding the levels of spend- ing. The specification of governmental spending targets, lower than their current levels, affects the corresponding activity levels of the remaining financial variables. The moderate increase in fees and charges seeks to compensate for savings in governmental spending, whilst there is reduction in almost all spending activities. A positive ele- ment is that the level of investment remains un- changed as also does the total level of borrowing. It is evident that the governmental scenario has a conservative tendency as its rationale is mainly focused on reducing the total level of spending by local authorities. This policy could be a result of adverse economic conditions or a result of a governmental policy which seeks to reduce ac- tivity levels of the local authorities. The diverse results of the two scenarios are considered as the initial conditions upon which the working party of the local authorities' finance could embark its negotiation process towards reaching a mutually agreed allocation plan. This process was pursued further by generating sensi- tivity analysis information of the optimal solution of the two scenarios. 5.4. Sensitivity analysis as an aid to negotiations The members of the working party were given information on the sensitivity of the two scenarios to changes in the level of global targets, and/or changes in the level of preferences in the objec- tive function of GoDEA. Particular emphasis was given on the tradeoffs between the optimal achievement of various controllable inputs/out- puts included in the analysis. The example chosen focuses on how the achievements on all global input/output targets change by changing the preferences over the min- imisation of deviations on the wages goal. The E+07 Drachmas 50 40 ~ . . . . . . . . . 0 10 20 30 40 50 60 70 80 90 100 Preferences over minimk~Cdon of wag~ deviatiork~ J * wages • invest o loans • services I Fig. 4. Implications of varying preferences on salaries on GoDEA solution. central government's scenario was chosen for this illustration and is given in Fig. 4. On the horizontal axis of Fig. 4 are listed the level of preferences over the minimisation of the positive deviations on personnel wages. The ver- tical axis contains information on the aggregate level of the four spending variables and how it is affected by the varying preferences over the personnel wages. The varying level of spending of each variable is represented with a corresponding piecewise equation. Increased preferences over the minimisation of overspending on wages has negative interaction with the level of spending on investments. In other words, reducing spending on wages will release extra resources to undertake investments. A weaker negative association can also be seen be- tween the savings on wages and the expenditure on services. It seems therefore that had the local authorities adopted a strategy towards reducing the level of spending on wages, this would affect positively their investment programmes and also the provision of services. As both investments and services have their own personnel require- ments, the proposed savings on wages for services indicate reallocation of activities and not reduc- tion to the level of employment. An extensive set of sensitivity results were requested by the panel of decision makers to facilitate the negotiations between the two groups, seeking a mutually agreed set of targets. 548 A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 Table 5. Reallocation of inputs-outputs among local authorities No. of local authorities Governmental scenario Local authorities scenario Reduction Increase Reduction Increase Governmental 31 16 6 42 Wages 35 12 3 44 Loans 20 17 18 29 Maintenance 23 24 19 28 Service expenses 29 18 7 40 Fees&charges 19 28 I1 36 Investments 28 19 10 37 5.5. Input~output reallocation among LAs Results on the achievement of the global input/output targets summarise the extent of in- put/output reallocation among individual LAs. The goal programming structure of GoDEA leaves open the option of increasing or decreasing the level of inputs/outputs of some of the LAs irrespective of the global target levels. Results of the number of LAs, at each planing scenario, with increased or decreased levels of inputs/out- puts are summarised in Table 5. Table 5 shows the resource implications on individual local authorities from the two planning scenarios. Each pair of columns shows the number of LAs that received increased/reduced funds from the corresponding planning scenario. The results show that the reallocation of inputs and outputs in GoDEA does not follow a pro rata pattern. In the local authorities scenario, where the global grant level is increased, some local authorities will have reduced funds com- pared to their observed levels. On the other hand, in the central government scenario, where the global level of central grants is reduced, there are 31 local authorities with increased levels of cen- tral grants gained at the expense of the 16 local authorities with reduced central grants. The selection of inputs and outputs to be real- located is made on the basis of the performance of the corresponding units as compared with their peer efficient local authorities. The reallocation is, undoubtedly, affected by the level of global targets and the preferences over their achieve- ment. More advanced reallocation scenarios can be pursued using GoDEA by allowing reallo- cation of demand characteristics among different local authorities. The latter can be implemented as part of a reorganisation plan that would affect the current size and boundaries of certain local authorities. 6. Conclusions This paper has sought to extend the traditional efficiency based orientation of data envelopment analysis models towards a combined use for re- source allocation and target setting. Resource al- location models, however, have more demanding structure in the sense that the interactions be- tween individual DMUs need to be taken into account and also the aggregate nature of re- sources for allocation needs to be encapsulated. The proposed formulation via the GoDEA model • links the allocation of resources with the es- timation of performance targets between individ- ual DMUs; • allows reallocation of resources and outputs among individual DMUs; • selects efficient technologies for individual DMUs in the light of the satisfaction of the global objectives of the system; and • incorporates decision makers' preferences from different levels of management. GoDEA can be employed for the generation of planning scenarios and also to aid the nego- tiation process between different levels of de- cision making. An implementation of the model was attempted seeking to support the reorganis- ation of the resource allocation system of local authorities in Greece. In general terms, GoDEA A.D. Athanassopoulos / European Journal of Operational Research 87 (1995) 535-550 549 was found an effective instrument of policy mak- ing beyond the strict optimisation form of the method. The planning objectives such as effi- ciency, effectiveness and equity were brought to- gether and the preferences of decision makers from the central and local government were intro- duced within the developed scenarios. This is what our experience with the application of GoDEA in a highly political environment has shown and, undoubtedly, more of these kinds of attempt should follow improving the implemen- tation of the methodology. Future research should also explore the contribution of DEA and resource allocation models to accommodate de- centralised forms of decision making and also to provide more explicit representation of the objec- tives of resource allocation as has been developed along the lines of the current paper and also in previous research by Athanassopoulos (1995). Appendix The mathematical programming model used to obtain the preferences of the two groups of de- cision makers used in the study is provided in (A.1). The model is based on the constraint re- gression analysis models developed by Charnes et al. (1986) and Sueyoshi (1994). Min eZ + eL RL (A.1) c + w ,e L ,e L subject to C C -I- w SL+eL- -eL =RL VL, c~C E lw l = lOO c~C w c free variables, -b e1_,eL ~0, where: S~. is the score of the L-th local authority on the c-th criterion used in the questionnaire. 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