Mathematical programming applications in educational planning

SO&I-ECOTI. Ph.

ski. Vol. 7, pp. 19-35 (1973). Pergamon Press. Printed in Great Britain

MATHEMATICAL PROGRAMMING APPLICATIONS IN EDUCATIONAL PLANNING JAMES F. MCNAMARA” Center for the Advanced Study of Educational Administration, Eugene, Oregon 97401

University of Oregon,

(Received 30 May 1972)

The specific aim of this article is to examine recent developments in the applications of mathematical programming techniques to problems encountered in educational planning. Applications are given for selected problems at national, state, regional and local levels of planning and the implications of this work for educational research methodology is noted. Special attention is given to applications at the microanalytic or school district level, since most applications in this domain are of recent origin and have been developed in disciplines other than education, e.g. industrial engineering, econometrics, public administration, business and operations research. Applications are discussed in light of their relationship to theoretical and empirical research on educational production functions. The final section contains some directions and implications for future research which are discussed in terms of recent developments in socio-economic and public sector planning and the emerging major research needs in educational policy planning. DURING the past few years there has been a significant increasing interest placed on an examination of the extent to which models developed in operations research, management science and econometrics might be used to improve the methodology and procedures currently employed in various areas of public sector planning and policy analysis. This type of inquiry has led some social scientists and public administrators to conclude that the problem-solving approach, inherent in economic theory and operations research, and structural-functional analysis, which is common in several social sciences, can be used collectively to develop better measures for socio-economic planning and resource allocation. While a synthesis of these developments in each of the several domains of public sector planning is in itself an important topic for planning specialists, our objective here is to focus on these developments within the educational sector. The specific aim of this article is to examine recent developments in the applications of mathematical programming techniques to various problems encountered in educational planning. In general, it can be said that research findings are science’s short-range benefits, but the method of inquiry is its long-range value. For this reason, the review of past applications provides some valuable insights into how mathematical programming techniques can be more effectively utilized in future educational planning efforts. The emphasis on future applications as well as an emphasis on current limitations and methodological procedures encountered in model development provide the major themes for each topic area.

* James F. McNamara is a research Associate at the Center for the Advanced Study of Educational Administration and an Associate Professor of Public Affairs and Administration in the Lila Acheson Wallace School of Community Service and Public Affairs at the University of Oregon. 19

20

JAMESF. MCNAMARA

The balance of the article is structured as follows. First, a survey of recent applications at the macroanalytic level is presented. This includes a review of applications at the national and state levels. Next, microanalytic or institutional applications are reviewed. This section is divided into two parts, one on university level planning and the other on school district level planning. Special attention is given to applications at the school district level since most applications in this domain are of recent origin and have not appeared in many of the previously published reviews. This is followed by a brief description of some criteria, stated in the form of questions, that can be used to evaluate models reported in the literature and also to analyze the general properties of optimization models that may be proposed in connection with some future planning or policy analysis. An outline of major problem areas in educational planning is included in the final section. This should be of particular value to planners with different disciplinary orientations, who are interested in the application of various analytical methods to emerging and unresolved issues in educational research and policy. Many of the applications not included in this review have been analyzed extensively in one or more of the surveys identified in the References. MACROANALYTIC

APPLICATIONS

The large amount of resources devoted to education and the increasing emphasis to design educational policy in relation to an overall set of objectives for economic and social development has resulted in the creation of comprehensive educational plans in virtually all the major nations of the world. The growth of actual planning operations has been paralleled by a rapid increase in the use of quantitative approaches designed to analyze basic economic characteristics of an educational system. These characteristics include its internal productive relationships and, more important, in light of macroanalytic models, the nature of the demand for its output in the economy. Since mathematical programming allows a planner to view an educational system as a set of input-output or production relationships, which can be controlled in a way that will optimize the use of scarce educational resources, it becomes a valuable technique to generate policy or “decision-oriented” information. Programming models with investment in education as a component of aggregate national investment have been reviewed [1, 21. Common to this type of macroanalytic model are the use of a dynamic programming approach with a planning framework of eight to ten years, and a simultaneous optimization of investments in education and real Indicative of this type of model are those capital for different sectors of the economy. recently developed [3, 41. In the Bowles model, the educational system of Northern Nigeria is simulated using dynamic linear programming model with an 8-yr planning horizon. The instrument variables include enrollments and resources used at the various educational levels (primary, secondary, etc,) in each time period. The objective function is used to maximize the present value of the contribution of various educational levels to future national income. Constraint categories reflect school policies regarding items such as (1) types of teachers, (2) student enrollments and transfer among the various types of schools, (3) school construction and current facility usage, (4) teacher recruitment or importation of foreign instructors, and (5) political or legal restrictions. Golladay has constructed a similar macro level model for Morocco in which the educational system is represented in the form of an intertemporal input-output model. The educational production functions (objective functions) are the conventional linear Leontief

Mathematical Programming Applications in Educational Planning

21

type and the economy is modeled as a modified, dynamic Leontief system. Other such models developed for the educational sector and the planning implications of this type of activity are reviewed in [5]. One obvious intrinsic property of these macroeconomic educational planning models is that they treat the educational system at its highest level of aggregation. Hence, within a single model, the objectives for primary, craft, technical and secondary schools in addition to universities are simultaneously optimized. Moreover, these models have been applied almost exclusively in developing countries where severe limitations are usually placed on the availability of funds for the educational sector. Also in these countries there seldom exists several discrete forms of educational legislation such as can be found in the United States where we currently have separate federal legislation for elementary and secondary schools, vocational education, higher education, manpower planning, etc. These considerations outlined above, coupled with the reality that education is basically a state responsibility (a decentralized system for control and management), explains in part why such aggregate models are seldom developed in this country. The position taken here, namely, that much can be learned from a careful examination of planning models and problem solutions derived in less-developed countries, has been recently documented by educational planners such as [6] and the participants at the recent U.S. Office of Education Conference on Operations Analysis [7]. In the United States there has been a recent interest shown in the use of mathematical programming models to analyze certain problems encountered in state level educational administration. This includes the models developed for vocational education planning [g-10] and for an investigation of alternatives to existing state level financial allocation formulae used to distribute funds to local school districts [ll, 121. These efforts to date are exploratory and their solutions have not yet been directly implemented. Section 123 of the new vocational education act (P.L. 90-576. The Vocational Education Amendments of 1968) dealing with new state plans makes it clear that each state will be expected to develop a state-local planning procedure that will assure the best use of funds in light of important training needs of all people as well as the requirements of employers. Based on the requirements of this legislation, it seems that those interested in state-level planning would be well advised to investigate the utility of mathematical programming and other forms of optimization models developed by research agencies such as the Organization for Economic Cooperation to analyze alternative human resource development policies [ 13-161. One major difficulty encountered in attempts to adapt macro models for future use in state-level planning is on the current ideology or position of many American educational planners toward the relative merits of the manpower requirements and social demand approaches to human resource development planning. Although numerous attempts have been made to clearly demonstrate the interrelationship of these two approaches, many planners and policy makers are reluctant to accept this position. In an effort to resolve this problem, some model builders have attempted to integrate several planning approaches in an explicit fashion within a single model. For example, Handa, in his recent econometric analysis of educational investment, has proposed a planning model that includes endogenous and exogenous variables based not only on an integration of the social demands and the manpower requirements approaches but also on the production processes within the educational system [ 171. The McNamara model provides a second illustration. He has shown that the problem of selecting vocational education

JAMESF. MCNAMARA

22

programs in a regional training center, where quotas are established for a set of partcipating school districts, can be structured to simultaneously optimize both labor market needs and individual student demands for training. The methodological problems dealing with model assumptions and mathematical limitations of objective functions or constraint sets in various programming models have been discussed extensively in the general econometrics and optimization literature. Although their importance cannot be minimized, they are not explicitly treated in this review. A second equally important and related type of methodological problem deals with the internal consistency and the measurement of educational planning variables that are subsequently used in optimizing models. The balance of our discussion on macro level applications in educational planning will focus on this consideration. Since all mathematical planning models are based on the explicit identification of variables and depend on some a priori knowledge of how this system of variables are functionally related, comments offered here also apply for models developed at the micro level. A major difficulty encountered in the development of general optimization models for the economy and education is the determination of functional forms of activities operating within the educational sector. Benard notes that the activity of the educational sector operates in at least four directions when it is viewed as an industry producing the knowledge required by future workers in the labor force [I]. These are: I. It provides pupils with the knowledge essential for the general or occupational skills they will later possess as members of the labor force (including teachers and research scientists). 2. It raises their cultural level and so influences the choices they will make and their abilities to absorb fresh knowledge during their working lives. 3. It develops scientific research within the universities themselves. 4. It helps to disseminate cultural, scientific and technical knowledge within the population as a whole through books and reviews, broadcasts, and the extramural activities of teachers. Given this orientation, the output of the educational sector is thus regarded as conSuch models do not allow the product of the sisting entirely of intermediate products. educational sector to be viewed as a flow into final personal consumption. In the absence of a theory of educational planning or development, the problem of analyzing the internal consistency of the relationships within a macro level optimization model of the educational system cannot be accomplished based only on criteria specific to educational systems. Hence, model builders must depend on a more abstract set of criteria such as that found within the economic development literature. To maintain consistency in the selection, measurement and functional form of variables used in the construction of multi-sector activity analysis models, Stone has suggested that the following seven classes of consistency should be analyzed [IS]. These are: 1. 2. 3. 4. 5. 6. 7.

Consistency Consistency Consistency Consistency Consistency Consistency Consistency

with with with with with with with

arithmetic identities. accounting identities. what we know about past behavior and technology. what we expect about future behavior and technology. transitional possibilities. all aspects of the problem. all our long-term aims.

Mathematical

Programming

Applications

in Educational Planning

23

Although dynamic programming has been used as a means to partially control for the problems of internal consistency outlined in (5) and (7) above, the major factor needed to develop a theory of educational planning to guide future model construction rests with the design of more precise educational production functions that address themselves to the problem of multidimensional outputs within educational systems. The positions taken here have been documented by a number of educational planners and are well summarized by Bowles [19]. He notes: In setting school policy and in long-range educational planning, knowledge of the educational production function is essential to efficient resource allocation. This is true whether the decision unit is pursuing the objective of growth or of equality, or a combination of these and other (perhaps noneconomic) goals. Without an estimate of the technology of education (the production function) the relationship between policies must be little more than guesswork (p. 12). Moreover, the school output is multidimensional, and the relative valuation of different outputs-say, mathematical competence as opposed to citizenship-differs among school districts. For this reason, technical inefficiency may result neither from inadvertence nor from the absence of optimizing behavior but rather from the conscious pursuit of objectives not adequately measured in any single index of school output (p. 17). Research on the development of production functions for educational systems can be generally categorized as either primarily theoretical or analytical in its orientation. Analytical studies differ from the other category in that they are empirically based and have resulted in the development of mathematical models, usually based on multiple linear regression. A recent and comprehensive overview of research findings in each category can be found in Copa [20]. He provides the reader with a summary of twenty-two major research efforts which form the basis for his proposed statistical model. Similar efforts have been reported and reviewed [3, 19, 21-231. In an earlier work Correa has proposed that two levels of educational production functions be established [24]. Micro level production functions should be determined to provide a framework to search for factors affecting the quality of schooling and the optimal use of resources within an organization. Macro level production functions should be used to analyze the structural and functional properties that arise when various educational organizations of a given social group are aggregated. Thomas [25], in a theoretical work on microanalytic levels of educational planning, describes production functions for administrators, psychologists, and economists so that three distinct types of input-output relationships in schools can be clarified. This is a particularly important distinction because it leads toward an integration of the research on the use of mathematical programming in learning theory, curriculum planning, human resource planning and administrative science as well as toward a more general concern, an economic theory of educational management. Although econometric models of production in education contain some methodological limitations since they are usually based on large cross-sectional statistical analysis, they have provided some empirical guidelines for the construction of objective functions and parameters. This approach has been demonstrated recently by Swanson [26]. Based on the results of two econometric models that developed descriptive input-output measures based on samples of fifty school districts in New York State and 119 districts in the northeastern

24

JAMES F. MCNAMARA

states, Swanson has developed a linear programming model designed to search for optimal resource combinations that maximize total benefits within a school district. The results obtained from this type of analysis should be viewed carefully, however, since it involves the use of parameters derived from probabilistics or stochastic analysis within a deterministic model. Sensitivity analysis or parametric programming techniques might be used to examine the effect on the optimal solution when these parameters are varied. Another analytical method that can be used to integrate theoretical and analytical approaches to educational production functions is goal programming, a technique used to derive operational objectives for resource allocation and target level planning in public administration [27, 281. Based on the use of linear or mathematical programming, goal programming is a method whereby the importance of each organizational objective is indicated not only by weights but also with absolute priority ordering. Goal programming represents an explicit or formal method for systematically relating the inputs and outputs used in nonmarket institutions to produce both consumer and investment goods. MICROANALYTlC

APPLICATIONS

In this section our concern is with the application of mathematical programming models at the micro or institutional level. The first part deals with model applications to planning problems encountered in university settings. We then turn toward recent applications within school districts. In general, universities have made far more progress in applying mathematical programming and other management science models to academic administration than have local school districts. This differential rate might be explained in part by the existence of departments of operations research, industrial engineering, computer science and management in universities. Major reviews of previous university level applications can be found in [2, 29, 301. University level planning In general, applications at the university level deal with eitherdepartmental management problems or with general university level planning. At the department level McCamley has used an activity analysis to determine the optimum output and activity levels of both a department within a college and for a hypothetical college composed of three departments [31]. His model contains forty-six different commodities (departmental services) in the activity vectors. Departmental outputs include degrees conferred, teaching to support students in other departments, three types of research activities and publications. Fox has used a linear programming model for allocating a given faculty among alternative teaching and research assignments [32]. Recursive and dynamic programming are used to extend this effort to compute optimal decisions over a sequence of years. A two-level decision model involving interaction between a dean and department chairmen in planning resource allocation is also provided by Fox. Briefly, this problem centers around the following organization posture. The objective functions for the various departments at one organizational level are all viewed by the dean as constraints, i.e. each comHowever, the dean also has an objective function which peting for part of his resources. involves maximization across all departments in his jurisdiction. This problem is handled by using a two-stage optimization model. Part of this research is also described in [33]. Turksen and Holtzman have recently developed mathematical programming models related to departmental levels in the organizational structure of a university [34]. Their programming formulations are in linear and quadratic integer forms. The formulations

Mathematical

Programming

Applications

in Educational Planning

2.5

are extended into dynamic programming for finite short-range planning purposes. Based on this and other previous work, they also have constructed an iterative dynamic programming model to facilitate short-term operational planning and forecasting [35]. Andrew has designed a linear programming model for matching available faculty in a department to courses that must be taught in a particular semester [36]. Faculty are assigned to scheduled courses based on measures of their effectiveness and teaching preference for each course offered. Flexibility for the decision-maker is built into the objective function by allowing c! to be the weight of teacher performance and (1 - c() the weight of the effectiveness criterion. Thus, if an administrator wished to place a weight of 0.35 on teacher preference, 0.65 would be the weight on teacher effectiveness in the system. This type of model could be easily adapted by a secondary school department. Geoffrion, Dyer and Feinberg have designed an interactive mathematical programming approach to a multi-criterion optimization problem within an academic department [37]. Six criterion functions are used simultaneously in an effort to model the allocation of faculty effort among three principal activities. These are formal teaching, departmental service duties such as administration or curriculum development, and other tasks such as research or student teaching. Since the aggregate objective function is not explicit, interaction on the part of a decision-maker is required at various stages in the solution algorithm. At the systems planning level, least cost decision rules for transferring library materials from primary to secondary storage have been developed by Mann using a dynamic programming model of a university library system [38]. Crandell has used linear programming to build a constrained choice model for student housing [39]. The purpose of this model is to aid in a university’s decisions regarding both construction of and subsidies for student housing. Graves and Thomas have used optimization models to analyze the problem of geographically allocating planned classroom spaces on a new campus [40]. Of particular value in their model is the utilization of different procedures in the sensitivity analysis other than the traditional forms such as studying the “right-hand side” vector or individual coefficients in the contraint matrix. Harden and Tcheng have developed and implemented a linear programming model designed to solve the classroom utilization problem at Illinois State University [41]. This operational model can beused to reexamine the scheduling problem when large annual expansions occur in full-time university enrolments. Vivekananthan has recently developed an educational research and development planning system that consists of a multistage model for formulating projects and a “reward” model for selecting projects [42]. Mathematical programming techniques would be used in the selection model to maximize expected value among any particular alternative combination of projects. Budgetary limitations are used to formulate a constraint set. Also explained are procedures to determine project utility, probability of a successful project completion, project cost, and total resources available. An optimization procedure for using information about private demand for higher education to help achieve objectives concerning the size and composition of total enrollment, given a fixed total subsidy for college students, has been developed and tested using California data by Hoenack [43]. Three separate but related objective functions are used to correspond with particular decision alternatives that may be maximized. These outcomes are subsequently related to subsidy and tuition proposals. Application of different mathematical programming techniques for university management have been developed in connection with the Generalized University Model Project at

26

JAMESF. MCNAMARA

the University of Texas [44,45]. Mathematical programming also has been used in a number of university management studies developed in connection with the Research Program in University Administration coordinated in the Office of the Vice President-Planning and Analysis, University of California [46-481. Research on the application of the state-space model for resource allocation in higher education at Michigan State University is described in [49]. The relationship of optimization models and simulation techniques for university planning are discussed in [50, 511. Basically, simulation models attempt to associate cause with effect and these models do not require the explicit objective functions needed for optimization models. Programming models, developed initially for university management, might be adapted for public elementary and secondary school management in larger school districts characterized by (I) a large set of diversified instructional facilities and resources, (2) a complex organizational hierarchy and (3) a multi-level program-planning-budgeting process. In each case, these models could be used to explicitly formulate a control process, or (without specific mathematical relationships) to evaluate current management procedures. School district planning Interest in mathematical programming models and other operations research techniques designed to increase the efficiency of a local school district management and policyformulation is a relatively new concern. This is especially true in areas of instructional policy formulation rather than in non-instructional problem areas such as school transportation and inventory control. For example, few mathematical programming models have been utilized to determine the gains of specific curricular contents for the individual student or to a total school program. A few exceptions found in the literature are reported here. Economists, who are concerned with this micro level of educational planning, usually develop programming models which assume that school systems are trying to maximize the net additional lifetime earnings of their graduates [52]. Their concerns center on relating earnings to the level of education completed, i.e. Grades 9, 10, 11, etc. Hence, differentiation of graduates by specific curricular programs are seldom offered. Illustrations of how linear programming can be used to decide between general or vocational education have been proposed [53, 541, but little evidence of their subsequent use is available. Shina, Gupta and Sisson have proposed two mathematical models in an attempt to relate student achievement in schools to resource allocations [55]. In these models, Lagrangian multiplier techniques are used to determine an optimal resource allocation strategy. Taft and Reisman have noted that recent major studies of curriculum have concentrated on the content and method of presentation but have not made significant strides toward systematic exploitation of the potential contribution made by the learning process of reinforcement through proper sequencing of subject matter (p. 926) [56]. They provide a curriculum development model using a heuristic algorithm in order to optimize the sequencing of subject matter. Sensitivity analysis and simulation are used to check different schedule combinations generated for particular levels of mastery. The authors suggest that such a model could be used at all levels of the educational system from elementary through graduate school, in highly academic as well as vocational training programs. Decisions about patterns for grouping students, assigning staff members and selecting programs are central to curriculum planning in school systems. Nystrand and Bertolart have observed that the most extensive work on coordinated planning has been in the area of master schedule building [57]. The University of Pittsburgh project report (see [SS])

Mathematical

Programming

Applications

in Educational Planning

21

included a number of algorithmic, linear programming and heuristic models for master scheduling. In this report Harding has shown that linear programming techniques for master schedule building can be used to determine “shadow price” information. These data constitute a means for evaluating the marginal returns for potential changes in physical plant, personnel resources, student curriculum demands and other instructional policies. Shapley et aE. [59] have used the transportation algorithm to solve the problem of utilizing idle classrooms in Los Angeles. Lawrie developed an integer linear programming model for the master schedule building [60]. Turksen and Holzman recently have shown how other optimization techniques, including branch and bound methods, can be used to solve scheduling problems where feasible scheduling spaces are represented by sets of bolean lattice points [61]. Conant has illustrated how a school district administrator might use linear programing techniques to analyze operational problems about the division of labor when nonprofessional teaching aides are employed to assist teachers [62]. His model is designed to focus on the following problems: (1) How many teachers and aides should be employed given relative salary costs? (2) How should staff be assigned to tasks in the instruction process? (3) What instructional and noninstructional work is needed from teachers and aides ? and (4) How can these selection, assignment and output problems be solved to get most effective labor services for a given salary budget ? Based on the use of actual data from the Portland School District, a hypothetical school situation is used to illustrate how an optimal task mix for teachers and aides can be derived. In vocational education, Persons, Leske and Copa have designed a linear programming model that can be used by instructors in a farm management curriculum to analyze actual problems dealing with animal food consumption and marketing analysis [63]. The model is part of an instructional package aimed at an integrated development and demonstration of innovations in the adult and secondary agricultural education curriculum. Bruno has used linear programming to develop an alternative approach to the fixed step salary schedule used by most school districts [64,65]. The model can incorporate many factors which are considered important by teacher unions, school boards, etc. in salary evaluation. Hence, it might be used as a tool for collective bargaining with teacher unions. Nine factors are used to determine a flexible salary schedule which permits overlaps in the established salary hierarchy, i.e. it would permit highly qualified teachers to receive higher salaries than low qualified administrators. Wasserman has shown that mathematical programming can be used to compile an education price index based on index number theory [66]. Contrasts between (1) linear programming and indifference curve procedures, and (2) the relationship between the deterministic nature of linear programming and the random factors of other statistical models having specified probability density functions are briefly explained. Sisson and the Government Studies Center have suggested how the simplex algorithm and integer mathematical programming might be used to generate appropriate utility indicators for Educational Program Planning and Budgeting Systems [67, 681. Since the landmark Supreme Court decision in 1954 invalidating school segregation, the problem of racial desegregation of school systems has received a great deal of attention on the part of school district administrators. As a result of this problem a series of mathematical programming models, based primarily on the transportation algorithm, have been constructed in an effort to determine official policies or courses of action leading to desegregation in school districts.

2x

JAMES F. MCNAMARA

Illustrative of this type of model is the one developed by Clarke and Surkis which solves the school desegregation problem as a multi-product distribution problem [69]. Given (I) the distribution of students in a community by race, (2) the location and capacity of each school, (3) the ethnic composition desired at each school and (4) the configuration of mass transportation lines in the community, their system searches for a plan of assignment to schools which (a) achieves the desired ethnic composition at each school, (b) requires no student to travel more than a specific number of minutes per day, and (c) minimizes the total daily student travel time. Other similar programming models include those developed by Heckman and Taylor [70], Lutz [71], Newton and Thomas [72] and the general model of Koenigsberg [73]. Ploughman rt a/. has developed an assignment program for the Bloomfield Hills School District (Michigan) [74], which provides the following information: (1) probable pupil population growth within the district, (2) desirable building sites selected from potential sites currently owned or available to the district, (3) preferred construction schedules for new schools and (4) the assignment of geographic areas within the district to particular schools. Stuart has developed a mathematical programming model to evaluate alternative urban plans and improvement programs [75]. Using the basic goals for the Model Cities Program that have been established by federal legislation, he designs a program-objective matrix having twenty items. These are divided into three basic groups: housing, employment-income-education and health-safety-environment. Programming models, designed to examine the interrelationship of these goals and the effect of specific budget allocations among each public sector, are formulated for a hypothetical neighborhood having a population of 100,000. Sensitivity analysis is used to derive various policy parameters for each of the three basic policy areas mentioned above. A final application illustrates the range of school problems to which mathematical programming can be applied. Lutz has used dynamic programming to construct a computerassisted cafeteria menu plan [76]. Based on the classic diet problem, a monthly school menu is produced which meets all of the legal requirements set forth in the U.S. Department of Agriculture school lunch program. Provisions also are made to allow for ethnic considerations in selecting food items of equal nutritional value and acquisition cost. A simulation model for computerized, selective school lunch menu planning also has been developed by Tanner and applied in the New Orleans Public Schools [77]. Recent development of mathematical programming models for local school district activities almost exclusively originates in disciplines other than education, i.e. industrial engineering, economics, business and operations research. Educators should continue However, the operational value of these models could to encourage this type of activity. be greatly enhanced if educational researchers and administrators would assist model builders to interpret the realistic nature of their model assumptions and properties in light of the organizational characteristics of schools. For example, an administrator in a school system, confronted with multiple demands and pressures from various constituencies and special interest groups within the community, is more likely to be primarily concerned with viewing costs and benefits in light of their short-term or more immediate effects. Hence, he would not be particularly enlightened by a model that assumes his major role in a school district is to design a curriculum that insures the greatest net lifetime earnings for each of his students. The administrative problem just briefly outlined here is the one Thomas has addressed himself to in the development of the three different micro level educational production functions

Mathematical

Programming

Applications

in Educational

Planning

29

[25]. Mathematical programming concepts, and possibly other models, could obviously be of direct help in this area as educational planners attempt to develop a micro level theory of the educational firm or a general theory for educational planning [78]. This obvious lack of a theoretical structure accounts for the fact that the most successful applications of mathematical programming and other optimization techniques at the school district level deal with noninstructional problems such as transportation, building location or inventory usage, where clearly defined objective functions can be identified. DECLSION

MODELS

One difficulty that is generally recognized as a significant factor contributing to the gap between the development of optimization (decision) models and their application to educational decision making is the serious inability of educators and operations research analysts to effectively communicate with each other. Contributing to this state of affairs is the obvious failure to realize that quantitative, computer-formed rationality is only one of the rationalities appropriate to a complex administrative problem. Political rationality, economic rationality, the valuing process or human relations concerns may be equally “rational” approaches to the same problem. Since the value of solutions appropriate depends on how adequatelythe model represents the real world problem,effective communications between the consultant and his client are essential and must precede the model building activities. In many cases, a problem clearly defined is half solved-or as Pythagoras “The beginning is half of the whole.” expressed it : Based on these considerations, the following statements in the form of questions should provide sufficient insight to analyze models reported in the literature and also to review the general properties of decision models that might be proposed in conjunction with some future educational planning or policy analysis. What is the single system to be modeled? Should a unique model be built for each subsystem ? Who are the decision-makers? What assumptions are made about the decision-maker who would use the model? Who is responsible for the generation of alternatives? How are they developed and by what criteria are they compared ? What are the goals, overall objectives or targets involved? What is their purpose? Are priorities or weights involved? What is their purpose? Are performance and effectiveness involved? How are these concepts defined and how are they measured? What is the pertinent time span for the construction of the model as well as for its implementation once it can be validated ? What supportive information systems are required for the model? What are the controllable variables? (For which decision-makers?) What are the uncontrollable variables? (For which decision-makers?) Can multiple criteria be translated to a single feature of merit? Does the concept of optimization vanish without a single criterion ? How would you implement the model in its intended environment? Are costs of translating an ideal model to a workable model in the real work ‘prohibitive’? How would you teach people to do the things that the model asks people to do? While this list is not exhaustive,it

is, on the other hand, sufficient to illustrate

the types

30

JAMESF. MCNAMARA

of considerations that must be included when models are to be judged as to their contribution toward solving real and immediate problems faced by educational administrators and planners. FUTURE

DIRECTIONS

Tn the next IO-15 years, new social, economic, political and technical forces will require substantial modifications of educational programs and services as well as require changes in the tools and techniques used to design and evaluate these proposed alternative courses of action. In general, the literature on future directions in educational planning is aimed not on predicting the future but on developing alternative futures. It places greater emphasis on understanding the relationships between the descriptions of feasible alternative futures of education and the design of policies necessary to implement them. The literature recognizes the need to expand existing forecasting methods to include approaches such as the creation of “future histories” by imaginative projections or scenarios. This interest in what is sometimes labeled educational futurism has resulted in a relatively new and more general educational planning literature. Major efforts in this domain include the research of The UCLA Institute of Government and Public Affairs [79], The Designing Education for the Future Project [80], The Educational Policy Center at Syracuse [81] and 1985 Committee of the National Conference of Professors of Educational Administration [82]. This latter reference contains a history of the developments in educational futurism and includes an annotated bibliography of 200 major references. The purpose of this final section is not to duplicate the excellent reviews of the educational planning literature mentioned above. Based on these sources, however, the objective is to outline some of the ways in which mathematical programming and other operations research models might be used in the immediate future to improve the effectiveness of educational planning. This objective already has been accomplished in part, since a number of research implications, based on present findings, have been included in several of the topic area reviews presented in the previous sections. However, before we return to this final objective, it seems important to provide those who are not familiar with the educational planning literature some insight into those emerging problem areas that are of primary concern to educational planners and policy-makers. One of the most comprehensive outlines of the general research questions which should be given special attention in the future research on educational planning is the work recently completed by the UNESCO International Institute for Educational Planning [83]. Based on the responses of more than forty institutions and a content study of current research issues and activities, they have organized the emerging research needs for the early 1970s into six major areas and fifteen research clusters. These classifications are given below. (A) Performance of educational systems. I. Clarifying educational objectives. 2. Measuring indirect and non-monetary

benefits

of education.

(B) Mutual adaptation of educational and economic systems. 3. Harmonizing the planning for manpower with the planning 4. Improving occupational-educational linkages. 5. Confronting the problem of the educated unemployed. 6. Contributing to the rural transformation.

for education.

Mathematical

Programming

Applications

in Educational Planning

(C) Internal effectiveness of educational systems. 7. Reducing wastage from drop-outs and repeaters. 8. Strengthening educational evaluation. 9. Individualizing instruction. 10. Establishing the pre-conditions for continuing innovation. (0)

(E) (F)

Resources employed in education. 11. Seeking supplementary sources of educational 12. Improving the supply and use of the teaching Lifelong learning. 13. Mapping organized

out-of-school

finance. force.

education.

Governance, planning and management of education. 14. Widening participation in educational governance. 15. Advancing educational planning and management.

The National Educational Finance Project recently has completed a nation-wide study of the financing of U.S. public education beginning with pre-first grade education, continuing through junior college, and including adult and continuing education, but excluding the financing of education provided in four year colleges and universities. Their research includes a nation-wide assessment of the dimensions of educational need [84, 851 which was developed as an operational guide for policy-makers whose primary task is long-range educational planning. This compendium should be one of the first items offered to socio-economic planners and operations research specialists now engaged in educational planning and interested in an overview of major policy issues and program dimensions within the educational sector. Sometimes a glance into the future is best attempted by considering lessons learned from the past. Along with numerous research reports, the author has received recently more than fifty replies from researchers (in various academic departments and private firms) who can be identified with the development of programming models in educational planning. A few of their comments and current concerns for the future direction of this activity are offered below. 1. Mathematical applications to management should stay clear of large general models and concentrate upon specific problem areas. The mistake that educators have made for years involves trying to solve huge problems that do not have sufficient operational characteristics or an adequate knowledge base from which feasible alternative strategies can be derived. 2. If the general area of operations research or decision-oriented research is to preserve in educational management, then the appropriate approach seems to be the consistent application of these techniques to small problems. In that way, we will efficiently solve the large problems and, hopefully, move toward a general theory of educational planning. 3. Distinctions between linear programming as it applies to developments in programmed instruction or to research efforts in operations research and optimization models for educational management should be made explicit to increase the efficiency of information retrieval languages such as ERIC. The generic term, mathematical programming, should be added to the current lists of descriptors found in these systems.

32

JAMES F. MCNAMARA 4.

Mathematical programming models can be used as one of the technical decisionmaking components of the broad-based program-planning-budgeting and management information systems now emerging in many educational organizations throughout the country.

This last point is of particular importance since more than two-thirds of the states have now moved toward adopting some form of PPBS to be used in both state and school district level planning. Based on a strong commitment to planning and a carefully designed implementation strategy, an innovation such as decision modeling can be utilized successfully in educational organizations to effect needed changes and improvements. To the model builder and the decision-maker interested in how mathematical programming can be used as an innovative tool to improve educational planning and to enlarge the domain of educational policy research, it might be mentioned that the risks of innovation were well understood by Machiavelli. He described them in “The Prince” as follows: “There is nothing more difficult to take in hand, more perilous to conduct, or more uncertain in its success than to take the lead in the introduction of a new order of things, because the innovator has for enemies all those who have done well under the old conditions and lukewarm defenders in those who may do well under the new.”

REFERENCES I. JEAN BENAKD, General optimization model for the economy and education, In: Organization for Economic Cooperation and Development. Mathematical Models in Educational Planning. OECD, Paris (1967). 2. KARL A. Fox and JATI K. SENC~UPTA, The specification of econometric models for planning educational systems: An appraisal of alternative approaches, Kyklos 21, 665-693 (1968). 3. SAMUEL S. Bow~ts, Planning Educational Systems fi)r Ecouonric Growth. Harvard University Press, Cambridge (1969). 4. FREDERICK L. GOLLADAY, A dynamic linear programming model for educational planning with application to Morocco, Doctoral Dissertation, Department of Economics, Northwestern University (1968). 5. RUSSELL G. DAVIS, On the development of educational models at Harvard CSED: an In: Education and Economic Growth, algebraic history of activity in one small place, (Ed. by RICHARD H. KRAFT). Educational Systems Development Center, The Florida State University, 1968. planning and American education, Planning and Changing 6. CHARLES S. BENSON, Systematic 2,4-I4 (1971). Proceedings of the symposium on 7. DAVID S. STOLLER and WILLIAM DORFMAN (Eds.), operations analysis of education, Socio-Ecorr. Plan. Sci. 2, 103-520 (1969). 8. JAMES F. MCNAMARA, A regional planning model for occupational education, SocioEcon. Plan. Sri. 5, 317-399 (1971). 9. CHARLES 0. HOPKINS, State-wide system of area vocational-technical training centers for Oklahoma, Doctoral Dissertation, Department of Agricultural Education, University of Oklahoma (1970). IO. CHARLES 0. HOPKINS, State-wide system of area vocational-technical training centers for Oklahoma, Research Coordinating Unit. Oklahoma State Universitv (1970). II. JAMES E. BRUNO, A mathematical pro&amming approach to school system-finance, SocioEcon. Plan. Sci. 3, l-12 (1969). to uniform expenditure reductions in multiple resource 12. JAMES E. BRUNO, An alternative state finance programs, Mgmt Sci. 17 B, 569-587 (1971). Models 13. ORGANIZATION FOR ECONOMIC COOPERATION AND DEVELOPMENT, Econometric in Education: Some Applications. OECD, Paris (1965). Models 14. ORGANIZATION FOR ECONOMIC COOPERATION ANI) DEVELOPMENT, Mathematical in Educational Planning. OECD, Paris (1967).

Mathematical 15.

17. 18. 19.

20. 21. 22.

23. 24. 25. 26.

27.

28.

Applications

33

in Educational Planning

ORGANIZATION

FOR ECONOMICCOOPERATION AND DEVELOPMENT, Eficiency in Resource OECD, Paris (1969). ORGANIZATIONFOR ECONOMICCOOPERATIONAND DEVELOPMENT,A simulation model of the educational system, Technical Report of the Center for Educational Research and Innovation, OECD, Paris (1970). M. L. HANDA,An econometric analysis of educational investment, Educational Planning Occasional Papers No. 2, Department of Educational Planning, The Ontario Institute for Studies in Education (1969). RICHARDSTONE, Consistent projections in multi-sector models, In: Activity Analysis in the Theory of Growth and Planning, Ed. by E. MALINVAUDand M. BACHARACH. St. Martin’s Press, New York (1967). SAMUELS. BOWLES,Towards an educational production function, In: Education, Income, and Human Capital, (Ed. by W. LEE HANSEN). Columbia University Press, New York (1970). GEORGEM. COPA, ldentifying inputs toward production function applications in education, Technical Report No. 4, Research Coordinating Unit for Vocational Education, University of Minnesota (1971). ELCHANANCOHN, Economic rationality in secondary schools, Planning and Changing 1, 166-174 (1970). HENRY M. LEVINE,The effect of different levels of expenditures on educational output, In: Economic Factors Affecting the Financing of Education, Vol. 2. National Educational Finance Project (1970). HENRY M. LEVINE,A new model of school effectiveness. Research and Development Memorandum No. 63, Center for Research and Development in Teaching, Stanford University (1970). HECTORCORREA,The Economics of Human Resources. North Holland, Amsterdam (1963). J. ALAN THOMAS,The Productive School: A Systems Analysis Approach to Educational Administration. John Wiley, New York (1971). AUSTIN D. SWANSON,Administrative accountability through cost-effectiveness analysis: a proposal, In: Emerging Patterns of Administrative Accountability, (Ed. by LESLEYH. BROWDER,JR.). McCutchan, Berkeley (1971). DONALD J. CLOUGH, Mathematical programming models of a quasi-independent subsystem of the Canadian Forces Manpower system, Working Paper No. 34, Department of Management Sciences, University of Waterloo (1970). EERO PITKANEN,Goal programming and operational objectives in public administration, Utilization

16.

Programming

in Education.

Swedish J. Econ. 72, 207-214 (1970). 29. KLAUS HUFNEK, Economics of higher education graphy, Socio-Eeorl. Plan. Sri. 2, 25-101 (1968). 30. JATI K. SENGUPTAand KARL A. Fox, Optimization Models. North Holland, Amsterdam, (1969).

and educational Techniques

planning:

in Quantitative

a biblioEconomic

31. FRANCISP. MCCAMLEY,Activity analysis models of educational institutions, Doctoral Dissertation, Department of Economics, Iowa State University (1968). 32. KARL A. Fox, Formulation of management science models for selected problems of college administration, U.S. Department of Health, Education, and Welfare, Office of Education, final Report Contract No. OEC-3-068058, Iowa State University (1967). 33. YAKIR PLESSNER,KARL Fox and SANYAL,On the allocation of resources in a university department, Metroeconornica 20, 256-271 (1968). 34. ISMAILB. TURKSENand ALBERT G. HOLZMAN,Short range planning for educational management, Ops Res. 18, Supplement 2, B-158 (1970). 35. ISMAILB. TURKSENand ALBERT G. HOLZMAN, Feasible sets for a class of scheduling problems, Department of Industrial Engineering, University of Toronto (1970). 36. GARY M. ANDREW,Operations analysis, Department of Management Science, University of Minnesota (1970). 37. A. M. GEOFFRION,DYERand FEINBERG,An interactive approach for multi-criterion optimization with an application to the operation of an academic department, Working Paper No. 176, Western Management Science Institute, University of California, Los Angeles (1971). 38. STUARTH. MANN, Dynamic management of library size, Ops Res. 18, Supplement 2, B-149 (1970). 39. ROBERTH. CRANDELL, A constrained choice model for student housing, Mgmt Sci. 16, B, 112-120 (1969).

JAMESF. M~NAMARA

34

40. ROBERTJ. GRAVESand WARRENH. THOMAS,A classroom location-allocation model for campus planning, Socio-Econ. Plan. Sci. 5, 191-204 (1971). 41. WARRENR. HAKDENand MIKE T. TCHENG,Classroom utilization by linear programming,

Technical Paper, Office of Academic Planning and Institutional Research, Illinois State University (1971). 42. P. S. VIVEKANANTHAN, The development of a research and development planning model in education, Center for Occupational Education, North Carolina State University at Raleigh (1971). 43. STEPHANA. HOENACK, The efficient allocation of subsidies to college students, Anr. Econ. Rev. 61, 302-311 (1971). 44. SUBRATASEN, Models of course-sectioning

45. 46.

47.

48.

49. 50.

51.

and faculty Assignment, In: University management models, Project GUM, Phase 111,Graduate School of Business, University of Texas at Austin (1969). M. SCHOEMAN,Class scheduling algorithm, In: University management models, Project GUM, Phaseill, GraduateSchoolof Business, Universityof Texasat Austin (1969). D. J. BARTHOLOMEW, A mathematical analysis of structural control in a graded manpower system, Paper P-3, Office of the Vice President-Planning and Analysis, University of California (1969). ROBERTD. SANDERSON, ‘The expansion of university facilities to accommodate increasing enrollments, Research Report No. 69-&Office of theVice President-Planning and Analysis, University of California (1969). GEORGE B. WEATHERSBY,Educational planning and decision making: rhe use of decision and control analysis, Paper P-6, Ofice of the Vice President-Planning and Analysis, University of California (I 970). RITA ZEMACH,A state-space model for resource allocation in higher education, IEEE Trans. Sys. Sri. and Cjaber. 4, 108-I I8 (1968). GEORGEB. WEATHERSBY and MILTONC. WEINSTEIN,A structural comparison of analytical models for university planning, Paper P-l 2, Office of the Vice President-Planning and Analysis, University of California (1970). DAVID HOPKINS, On the use of large-scale simulation models for university planning,

Rev. Educ. Res. 41, 467-478 (1971). 52. A. G. HOLTMAN,Linear programming and the value of an input to a local public school system, Public Finatzce 23, 429-440 (1968). 53. HECTOR CORREA, Optimum choice between general and vocational education, KvXlus

18, 700-704 (1964).

54. A. J. CORAZZINIand E. BAKTELL,Problems of programming: an optimum choice between general and vocational education, Kyklos 18, 107-l 13 (1965).

55. BANI, K. SHINA,S. GUPTA and R. SISSON,Toward aggrcg::tc models ofeducational systems Socio-Econ. Plan. Sci. 3, 25-36 (1969). 56. MARTIN1. TAFTand ARNOLDREISMAN,Toward better curricula through computer selected sequencing of subject matter, Mgrnf Sci. 13, 926945 (1967). 57. RAPHAEL0. NYSTRANDand FREDEKICLBERIUL~KT.Strategies for allocating human and material resources, Rau. Edtrc. Res. 37, 448-468 ( 1967). 58. ROBERT E. HARDING, The linear programming approach to master time schedule generation in education, in: Optimal Schedttlin~ in Educational Institutions, (Ed. by A. G. Holtzman and W. R. Turkes). U.S. Department of Health, Education, and Welfare, Officeof Education,CooperativeResearchProjectNo. 1323, University of Pittsburgh(l964). 59. R. P. SHAPLEY,D. FULKERSON,A. H~V~I.ICI( and D. WLII~LK,A ::.ansportation program for filling idle classrooms in Los Angeles. P-3 405, The RAND Corporation, Santa Monica, California (1966). 60. N. L. LAWRIE,An integer linear programming model of a school time-tabling problems, Computer J. 12, 307-316 (1969). 61. JSMA~LB. TURKSENand ALBERTG. HOLZMAN,Micro Level Resource Allocation for Universities, Ops Res. 18, Supplement I, B-39 (1970).

Models

62. EATONH. CONANT,A cost effectiveness study of employment of nonprofessional teaching aides in public schools, U.S. Department of Health, Education and Welfare, O&e of Education Final Reuort. Proiect No. OE-S-0481. institute of Industrial and Labour Relations, University of Oregon (1971). 63. EDGAR PERSONS,GARY W. LESKEand GEORGAA. COPA, Development and Demonstration of innovations in adult agriculture education, U.S. Department of Health, Education, and Welfare, Office of Education Final Report, Project No. O.E. 7-0758, Department of Agricultural Education, University of Minnesota (1970).

Mathematical

Programming

Applications

in Educational Planning

64. JAMES E. BKUNO, Using iinear programming salary evaluation models in collective bargaining negotiations with teacher unions, Socio-Econ. Plan. Sci. 3, 1-12 (1969). 65. JAMESE. BRUNO, Compensation of school district personnel, Mgmr Sci. 17, B, 386-398 (19711. Edacution Price and eaantity Indexes. Syracuse University Press, 66. &I&VI WASSERMAN, Syracuse (1963). 61. RICER L. &SON, The project concept in planning, programming, and budgeting, SocioEcon. Plan. Sci. 4, 239-261 (1970). budgeting system 68. GOVERNMENTS&DIES CENTER, Education-planning-programming procedures manual for school districts, Version 2, Model 1, Fels Institute, University of Pennsylvania (1969). 69. S. CLARKEand J. SURKIS, An operations research approach to racial desegregation of school systems, Socio-Econ. Plan. Sci. 1, 259-272 (1968). 70. LEILAB. HECKMANand HOWARDM. TAYLOR,School rezoning to achieve racial balance: a linear programming approach, Socio-Econ. Plan. Sci. 3, 127-133 (1969). 71. RAYMONDP. LUTZ, A systems study of school desegregation, Research Report R-70-2, School of Industrial Engineering, University of Oklahoma (1970). 72. RITA M. NEWTONand WARRENH. THOMAS,Design of school bus routes by computer, Socio-Econ. Plan. Sci. 3, 75-85 (1969). Mathematical analysis applied to school attendance areas, Socio73. ERNESTKOENIGSBERG;, Econ. Plun. Sci. 1, 465475 (1968). An assignment program to establish school attendance boundaries and 74. T. PLOUC;HMAN, forecast construction needs, Socio-Econ. Plan. Sci. 1, 243-258 (1968). 75. DARWIN G. STUART, Urban improvement programming models, Socio-Econ. Plan. Sci. 4, 217.-238 (1970).

76. RAYMONDP. LUTZ, Taking the heat off the school lunchroom, a paper presented at the Annual Conference of the American Institute of Industrial Engineers (1968). 77. CHARLESK. TANNER,An automated simulation vehicle for school business administration accentuating computerized selective school lunch menu planning, Doctoral Dissertation, Department of Educational Administration, The Florida State University (1968). 78. BARRYN. SEIGEL,Towards a theory of the educational firm, Technical Report, Center for the Advanced Study of Educational Administration, University of Oregon (1966). 79. WERNERZ. HIRSCH,Inventing Education for the Future. Chandler, San Francisco (1967). so. Eooaa L. MORPHET, (Ed.), Designing Edacation fbr the Future, Vols. 1-8. Citation Press, New York (1969). Xl. MICHAELMARIEN,Essential reading for the future of education: a selected and critically annotated bibliography, Educational Policy Research Center, Syracuse University Research Corporation (1970). 82. WALTERG. HACK, Educational Futurism 1985: Challenges for Schools and Their Administrators. McCutchan, Berkeley (1971). 83. WII.I.IAMJ. PLATT,Research for EducationalPlanning: Notes on Emerging Needs. UNESCO International Institute for Educational Planning, Paris (1970). 84. ROE L. JOHNS, KERN ALEXANDER and RICHARD ROSSMILLBR(Eds.). Dimensions of Educational Need. National Educational Finance Project, Gainesville, Florida (1969). 85. ROE L. JOHNS, KERN ALEXANDER and K. FOREIS JORDAN (Eds.). Planning to Fiuance Education. National Educational Finance Prqject, Gainesville, Florida (1971).

35