Using linear programming salary evaluation models in collective bargaining negotiations with teacher unions

Socio-Econ.

Plan.

Sci.

USING MODELS

Vol.

3, pp.

103-I

LINEAR

17 (1969).

Perpamon Press.Printedin GreatBritain

PROGRAMMING

IN COLLECTIVE WITH

SALARY

BARGAINING

TEACHER JAMES EDWARD

EVALUATION NEGOTIATIONS

UNIONS BRUNO

ASSISTANT PROFESSOR

University of California at Los Angeles

The widespread incidence of teacher strikes, usually associated with demands for higher salaries, has in many school districts throughout the country, established the need to develop effective salary evaluation schemes for school district personnel. Presently used methods of determining salary, such as the fixed step salary schedule, are not capable of considering multiple factors in salary evaluation. The purpose of this paper is to formulate, then apply, a linear programming model to determine a school district salary schedule. The model is capable of incorporating all factors which are considered important by teacher unions, school board, etc., in the salary evaluation. In addition, certain union demands or conditions reached as a result of arbitration can be included in the model. Finally limits placed upon school district resources available to support the salary structure can be considered. Essentially, the model can be used by a school district to determine a logical, internally consistent salary schedule which meets specific union demands at minimum school district cost. Finally the use of the model can give school districts the added flexibility of developing salary schedules which permit overlaps in the established salary hierarchies, e.g, it would permit highly qualified teachers to receive higher salaries than low qualified administrations.

INTRODUCTION IN MANY

school districts

areas, teacher

unions

throughout the country, especially those located in large urban and associations are demanding a larger voice in the determination

of salaries [I]. The widespread incidence of teacher strikes over the past few years underscores the growing seriousness of the problem. Essentially teacher union demands are multi-dimensional in nature. General or across-the-board increases in salary, and additional monetary consideration for other factors affecting the instructional process besides experience are two such demands. Developing a salary schedule which permits overlaps in the school district salary hierarchy, e.g., a salary schedule in which a highly qualified teacher could receive a larger salary than a low qualified administrator, is another demand of some teacher unions and associations. Finally teachers seem to desire the development of a salary schedule which is logical and internally consistent, with each member of the organization receiving a fair salary in relation to every other member of the organization. The present fixed step salary schedule, commonly used by most school districts, due to its inherent rigidity and inability to consider multiple factors in salary evaluation is inadequate for meeting these types of demands and specifications [2]. School boards in many districts have indicated a need for the development of a method which could be used as a mechanism in arbitration proceedings with teacher groups. The model would have to simultaneously consider all union demands, as well as the resources available to the school district to 103

J~ES

104

EOWARD

BRUNO

support salaries. This latter criterion is an extremely important requirement for developing an effective salary schedule for school district personnel, since the constraints placed upon school district resources usually determine the final salary structure. To develop a model which 1. considers any and all factors which teacher groups would desire to have included in the salary evaluation, 2. maintains the organizational salary hierarchy, but permits the overlaps in salaries between hierarchies, 3. considers the resources available in the school district to support the salary structure, 4. generates a salary schedule which is logical and internally consistent and determines the salary of each individual in relation to other members of the organization, is. obviously a very complex problem. Multiple regression analysis, frequently used in business and industry, for problems in wage salary evaluation, is not appropriate to this problem because constraints cannot be considered and some of the factors to be included in the salary evaluation cannot be historically correlated to salary. Fortunately, there are techniques of operations research, namely linear programming, which could be of important use to school districts for aiding in salary schedules. Recently linear programming has been suggested and applied to wage salary evaluation in business and industry [3, 41. The purpose of this paper is to develop, then apply a linear programming model to a school district salary structure. The objective or criterion of effectiveness used in the study is to develop an internally consistent salary schedule which satisfies certain specified union demands at minimum cost to the school district. DEVELOPMENT

OF

THE

SALARY

EVALUATION

MODEL

Preliminary considerations

Before developing an effective salary scheme for school-district personnel, the various criteria used in the scheme have to be discussed and agreed upon by school administrators, the PTA, the school board, and teacher groups. The essential questions to be answered would include the following: 1. What are the basic objectives of the school district (e.g., preparation for college, citizenship, social adjustment, etc.)? 2. What are the primary job functions or job classifications needed by the school district to attain these objectives (e.g., administrators, teachers, teacher aides)? 3. What factors-i.e., training and/or qualifications-are considered necessary for each of the job functions in the district (e.g., education, responsibility, subjectmatter training) ? 4. What is the hierarchial salary structure in the school district (e.g., will administrators be paid more than teachers, who will be paid more than teacher aides, etc.)? 5. What characteristics, in decreasing order of relative importance, constitute each of the factors in (3) above (e.g., education characteristics would be Ph.D., M.A., B.A.)? Since the essential characteristics of any job-evaluation scheme is consistency, it should be stressed that all parties concerned with teacher salaries (school board, teacher groups, PTA, administration) be included in these preliminary discussions.

Linear Programming

Salary Evaluation in Negotiations

with Teacher Unions

105

Devetoping the model Note that the linear-programming

salary-schedule evaluation scheme to be described later is not limited by the number of factors or functions which might be considered in Essentially, the linear-programming approach classifying school district personnel. calculates relative weights for each factor, such that the ranking of job functions (positions or job classifications) by salary corresponds to the ranking of that function in the schooldistrict salary hierarchy. The resulting analysis and solution of the model then provides the school administration with some measure of the relative importance of each factor used in the salary scheme. There are six phases to the development of a salary scheme by the use of linear programming: I. 2. 3. 4. 5.

Specification of the organizational salary hierarchy Determination of the factors to be included in the salary evaluation Development of a relative rating system for the characteristics in each factor Determination of the relative weights for each factor in the &valuation Determination of the salaries for the miscellaneous positions in the hierarchy after the key positions have been determined 6. Evaluation of positions and salaries of individuals within the system.

The application of these phases to the school-district job/salary-evaluation model is described below. Specification of the organizational salary hierarchy. The job functions or classifications which correspond to the school district salary hierarchy and relate to the goals and objectives of the school district can be viewed in many ways. This paper will consider the salary hierarchy from the framework of the individual’s responsibility within the job classification. These classifications would include responsibilities which are district wide, schoolwide, department wide, class wide, student. This roughly corresponds to superintendents, principals, department heads, teachers and teacher aides. Determination offactors to be included. The factors to be included in the analysis can usually be determined by an initial survey of the personnel in the school district who are involved in the evaluation. This is followed by several meetings at which all the factors to be used are agreed upon. Some of the factors used will probably reflect compromise situations, but this is not a major problem to the evaluation scheme. One important result of this phase of the study will be the improvement of ambiguous or poorly written position descriptions. Development of a rating system for characteristics in each factor. A relative rating for each characteristics of each function must be defined. For example, the characteristics in education might be ranked as follows: Characteristics Ph.D. or Ed.D. M.A. or MSc. M.Ed. B.A. or B&z. A.A. H.S. Diploma

Relative rating 5 4 3 2 1

0

106

JAMES EDWARD

BRUNO

Education is a relatively objective factor, since each characteristic is directly measurable by the completion of a college degree. A more subjective factor might be the difficulty of the learning situation of the school within the school district. A suggested rating scheme for this factor would be the following:

Characteristic Difficult; disadvantaged area Medium or averaged difficulty Not difficult; culturally advantaged

Relative rating 3 2 1

This rating scheme might present problems in schools districts. Since the Federal Government has developed indices for determining whether a given school is culturally disadvantaged, this type of index could be used. Other partial measures of the difficulty of the learning situation could include percentage of dropouts, percentage of minority groups, discipline or police records of the students, number of teachers requesting transfers, etc. A dichotomous classification (difficult-not difficult) could also be used for this factor. An important concept to remember is that a factor with too few ratings or characteristics will not sufficiently discriminate among deficiencies in ability whereas one with too many will result in ambiguity. The establishment of the ratings for each characteristic of each function is usually based upon the job description and the mutually agreed-upon order of relative importance of the characteristics to the particular factor. Determination of relative weights for each factor. The determination of the weights to be assigned to each factor is accomplished by means of the model. The model will yield a point system for a number of key positions in the school district. A key position might be considered to be one for which a pay differential has been established, such as between administration and teachers. These key positions usually correspond to the different functions of the personnel in the school district hierarchy. The values for the rest of the positions in the school district will be determined by the bounds established by these key positions. Determination of relative ranking of other positions in the school district hierarchy. The relative weights, determined by the solution of the model, are used to evaluate and place in a logical and internally consistent manner, the remaining positions in the school district. If the salaries for certain positions seem out of place with established school district policies in terms of salary, then either the mathematical formulation of the evaluation scheme must be revised, with more key positions included in the model, or else a significant factor (or factors) has been omitted in the analysis of the position. Evaluation of individual positions and salaries. The primary benefit of the proposed salary-evaluation scheme for school districts is the effective evaluation of the certified school district personnel-their positions in the hierarchy of the district, and their salaries.

Linear Programming

Additional

Salary Evaluation in Negotiations

with Teacher Unions

107

refinements

In addition to the essential phases, the school district may desire to include the following additional refinements in its wage salary model. I. Establishment- of the relative importance of the factors themselves.

2. Investigation of other environmental 3. Determination effectiveness.

constraints, especially budgetary.

or development of sophisticated objective functions or measures of

A school district may wish to place heavier emphasis upon a factor such as the difficulty of the learning situation or subject-matter preparation. The establishment of the relative importance of factors or combinations of factors can be established by an ordering of them by a relative rating scheme in the model. This procedure would be very similar to the relative ordering of characteristics within a factor. Summary

of procedures for developing the model

The procedures for developing the salary-evaluation be summarized as follows:

model for a school district may

I. Identification of the school-district personnel by function or position in the schooldistrict salary hierarchy. 2. Identification of those factors which contribute to the performance of the personnel in each function or position in the school-district organization. 3. Identification of the descriptors ordering by relative importance.

or characteristics

within each factor, with an

4. Formulation of the mathematical equations which represent the theoretically lowest- and highest-paid school-district personnel for each function in the salary hierarchy. 5. Inclusion in the mode1 of other environmental constraints, such as those reflecting the “financial environment” of the school district and the various interposition and intraposition percentage spreads in salary. 6. Determination of the objective functions or measures of effectiveness to be employed by the school district in the evaluation scheme. DERIVATION

Identification

OF THE

LINEAR

PROGRAMMING

MODEL

of personnel by function

In this illustrative application of the model, five school-district functions (job classifications) were considered. These were in decreasing order of responsibility and salary. I. Superintendent 2. Administrator 3. Department

(principals, vice principals, etc.) Heads

4. Teachers 5. Teacher aides and other teacher assistance personnel such as lab assistants and readers.

108

JAMES EDWARD BRUNO

Definition of relevant factors withinfunctions

Each of the school-district functions included in the model was evaluated according to nine factors. These factors although arbitrary are considered reasonable for illustrative purposes. 1. The type of area in which the school is located, e.g., the relative difficulty of the educational situation as determined by the teaching environment (X,). 2. The subject-matter area being taught, in terms of demand compared to available supply of teachers (X2). For example, high-demand areas would be English, mathematics, and science; average-demand areas would be Latin and French; low-demand areas would be physical education for men and social studies. For administrators a high priority skill might be a systems analyst, computer specialist, etc. ’ 3. Supervisory responsibilities of the personnel, in terms of the area of responsibility (X3)* 4. The highest academic degree attained by the individual at the time of the evaluation (X4). 5. The total work experience of the individual within the district, plus years worked in previous organizations or school districts (X5). 6. Special distinctions or awards earned by individuals, such as Phi Beta Kappa administrative or teaching awards (X,). 7. The number of college units or semester hours completed beyond the highest degree attained (X,). 8. The number of in-service-hours credits obtained per year by the teacher or administrator in programs of professional development (X,). 9. The relative additional workload corresponding to the individual’s job classifications within the organization (X,). The above mentioned set of factors should not be considered complete. Depending upon the objectives of the school district and compromise agreements reached with teacher unions, PTA groups, at school board, etc., other factors might also be considered. The linear programming approach is unique because there is virtually no limit to the number of factors which may be considered for the evaluation of salaries. For illustrative purposes only this study will limit itself to the nine abovementioned factors. Specification of hierarchial constraints

Once the factors, their characteristics, and the relative ratings of the characteristics are specified, it is possible to describe each function in the organization hierarchy by means of two equations; one represents the most theoretically highly qualified person for the function and hence the highest salary, and the other represents the theoretically lowest qualifications, hence the lowest salary. In generalized mathematical terms, the equations representing the highest (hj) and lowest (0,) levels within each function j take the following form in the model:

Linear Programming Salary Evaluation in Negotiations with Teacher Unions

109

where : %i2bi cc, = the highest rated characteristics

Pi = Xi ~j bj ,I

= = = =

associated with the factors appropriate to function j in the school district the lowest rated characteristics associated with the factors appropriate to function j in the school district the factors associated with function j the theoretically highest or maximum salary to be paid within function j the theoretically lowest or minimum salary to be paid within function j the number of factors considered in the evaluation scheme: i = 1, . . . n.

Dollar Differences between the highest salaries in each of the school districts job classification can be represented by the following equation . Aj+l-Lj
where

I Aj+

highest salary in job classification j+ 1 1 = the theoretically Aj = the theoretically highest salary in the next lower job classification j 6 = the specified dollar difference in salary.

Percentage overlaps in salary between job classifications (e.g., the lowest salary of the higher classification to the highest salary of the lower classification) can be controlled in the model by means of the following equation 6j>“ij+, where Rj+I

= the highest salary in job classification j+ 1 aj = the lowest salary in the next lowest job classification j o = the desired percentage overlap in salary between job classification j and j+ 1.

The percentage relationships between the theoretically highest and lowest salaries in each job classification or function can be mathematically expressed as: Oj2YAj

where aj = the lowest salary for job classification j lj = the highest salary in job classification j y = the percentage spread in salary for job classification j. Specification of budgetary constraints Finally, the budgetary constraints which would affect the entire salary structure can be incorporated in the model by means of the following equation where lril-= Xi = jlin = $ =

the number of certified employees having characteristic k of factor i factor i used in the evaluation scheme the relative rating given to characteristic k of factor i the total amount of school district funds available for distribution for certified: personnel salaries.

JAMESEOWAROBRUNO

110

Determination of the objective fllnction

In this study the minimization of school district costs objective function or specified criterion of effectiveness of the situation in which salary negotiations with teacher unions are would like to determine, at minimum cost to the district, a salary schedule which satisfies these demands.

ILLUSTRATIVE

APPLICATION

(‘%*)will be considered the model. This would reflect a in progress. School officials logical internally consistent

OF THE

MODEL

Inputs to the model

In order to validate the model and test its effectiveness in meeting certain desired objectives, it was applied to a school district salary structure. Table 1 lists the nine factors included in the salary model, thevariable representing this factor, the de.scriptive characteristic of each factor in decreasing relative importance, the number of school district personnel possessing each characteristic, the relative weight for each characteristic and finally the total relative weight for each factor. Assume the school district salary schedule applied to 1386 personnel which included a superintendent, 30 administrators, 90 department heads, I200 teachers, and 60 teacher aides.

Summary of the model used in the stud) Minimize ‘I’ Subject to Organizatiorlnl Hierarchy Constraints

Superintendent (Function 1) Administrator (Function 2) Departmenl Head (Function 3) Teacher (Function 4) Teacher Aide (Function 5)

3X,+3X,+7X,+5X,+7Xj+2X,+5x,+5X,+5XB+5X~<& X,+

X,+6X3+3X,+

3X, +3X, +5X, +5X,+ Xi+

X,+

X,+

X,+

X,+jX,>a,

7X, +2X, +5X, +5X* +4X, I A2

Xz+4X,+3X,+

Xj+

X6+

XT+

X*+4X,>a*

X2+3X3+2X++

Xj+

X6+

XT+ X,+3X,>a,

x,+2x3+2x,+

3X,+3X1+ xi+

X*+

X,+

x,+

x,+

x,+2x,20,

X,+2X,+7X5+2X,+5X,+5X,+ xJ+

x,+

Xj+

X6+

Percentage salary spreads within each job classification o,20.802.,

(4)

(6) (7)

3X,+3X2+2X3+5X,+7X,+2X~+5X,+2XgIi, X,+

(3)

(5)

3X,+3Xl+3X3+5X,+7Xs+2X6+5X,+5X,+5XB+3Xg<& Xi+

(1) (2)

XT+ X8+

Xpj& Xg2Cj

(8) (9) (10)

(11)

62 2 0.7522

(12)

n3 2 0.6Oi,

(13)

0,20.551,

(14)

Qj 2 0.5oi,

(15)

Linear Programming

Salary Evaluation in Negotiations

with Teacher Dnions

111

TABLE 1. FACTORYINCLUDEDIN THE MODEL*

Factor

Relative weight and characteristics

Variables

Learning environment

Xl

Number of employees possessing this characteristic

Weighted total for each characteristic

220

660 2332 0 2992 708 2030 135 2873 7 30 25 100 270 2400 60 2892 100 480 3300 2200 46 6126 112 600 1500 1600 1500 100 20 5432 72 1350 1422 3000 2000 600 100 36 5736 3000 2400 300 112 30 5842 30 120 270 2400 60 2880

3 Difficult 2 Medium 1 Easy

1166 0 236 1015 135

Subject matter or special skills

x”_

3 High Priority 2 Medium 1 Low Priority

Supervisory responsibility

X3

7 6 5 4 3 2 1

Single district wide District wide Simple school wide School wide Department wide Class wide Student

2 5 5 z 1200 60

Highest academic degree attained

X4

5 Ph.D. or Ed.d. 4 M.A. 3 M.Ed. 2 B.A. 1 A.A.

Work experience

X5

7 6 5 4 3 2 I

Special awards and distinctions

X8

2 with 1 without

College credits completed in addition to degree

X7

5 28units 4 22-28 ,( 3 15-21 ,, 2 S-14 ,, 1 O-7 ,,

600 500 200 50 36

In service units completed

X*

5 414 314 3 21-30 2 11-20 1 O-10

z 100 56 30

Relative additional workload in the hierarchy

X9

12years IO-12 ,, 8-10 ,, 6-a,, 4-6.3 2-4 ,, O-2 1,

5 District wide 4 School Wide 3 Department Wide 2 Class Wide 1 Student

20 120 1100 1100 46 16 100 300 400 500 50 20 36 1350

6 30 90 1200 60

* Notice each characteristic within each factor can be weighted. For example a year of experience for an administrator might be weighted twice that for a teacher. This study, however, will assume equal weights for each characteristic in each factor for all job functions in a school district. B

112

JAMESEDWARD BRUKO

Percentage

salary

overlaps

Ivlinirt~un~~lolltrr spreads

between job classifications

between

1,20.950,

(16)

A, 2 0.9001

(17)

A,20.80a,

(18)

A52 0.80a,

(19)

the highest salaries A, -1223000

of each job classification (20)

II,--i,23000

(21)

A,--&>

(22)

1500

I,--Aj21500 Scliool district

resources available to support 2992X, +2873X2+2892X3+6126X;I+5432Xs+

(23)

the salary structure* 1422X,+5736X,

+5842X 8-r’ 2880X,
bourltls placed

(24) upon the value of each factor 2501X,11000

in the model (25)

200Ix,I1000

(26)

lOO
(27)

lOO~x,<2ooo

(28)

505x,<

500

2501X,11000

(29) (30)

501x,<

300

(31)

2o
100

(32)

1001X,<2500

(33)

In the above model, 5 hierarchies or job classifications are specified with (1, 3, 5, 7, 9) representing the theoretically highest salary in the hierarchy and (2, 4, 6, 8, 10) the theoretically lowest salaries. These equations can be thought of as representing salaries of the theoretically most highly qualified and the theoretically least qualified individual in each functional classification in the school district. The spread in salaries within each * Notice if school districts were desirous of planning next year’s salary schedule and had openings for new personnel, the characteristics associated with factors 1,2, 3, and 9 would be known while the characteristics of factors 4, 5, 6, 7, 8, would be unknown. School administrators might assume that these new personnel have the highest characteristics for these factors (PhD, special awards, etc.). This procedure would insure that the amount of school district funds necessary to support the salary structure would not be exceeded. If a school district wanted to transform its present salary scheme to one based upon linear programming, then lower bounds constraints might be placed upon each individual’s salary to insure no one receives less salary.

Linear

ProgrammingSalary Evaluation in Negotiations with Teacher Unions

113

classification is controlled by the percentage relationships specified in (16-19). Spreads between the highest salaries among job classifications are controlled by means of (20-23). Overlaps in salaries (highly qualified teachers to receive higher pay than depa~ment heads) are controlled by the percentage relationships specified in (16-19) The resources available to a school district for salaries determine the salary schedule. These resources are considered in the mode1 by means of constraint (24). Finally upper and lower bounds are placed upon each of the factors constituting the salary schedule. This procedure insures that no single factor accounts for more than a certain part of the individual’s salary. These bounds are specified in (26-34). Results

of the stud]

The following table summarizes the resulting salary schedule obtained by the application of the wage salary model to this particufar school district. TABLE 2, VALUE FOR EACH FACTOR IN TRE SALARY EVALUATION SCHEME

Description

Value

Difficulty of Learning Environment Subject IMatter Priority Supervisory Level Highest Academic Degree Work Experience Special Awards or Distinctions College Credits Jn-service Units Additional Workload

5250.00 5527.37 6100.00 s100.00 550.00 S1000.00 SSO.00 S20.00 $2402.63

TABLE 3. OPVMAL SALARYSCHfDULE* Function

Function

1 Superintendents Assistants 2 Principals Assistants 3 Department Head

Function

4

Function

TeacheF

Function 5 ‘I’eaIdherAide Objective Fun&&

(highest) (lowest) (highest) (lowest) (highest) (lowest)

515,575 $14,860 $15,642 $12,207 s;13,564 $8,429

(highest) (lowest)

510,537 $7,102

(highest) (lowest) VI Total District Cost

-

57,434 s4,soo S11,916,248

+ Solving the modef for different objective functions (criteria of effectiveness) would, of course, yield different values for the factor weights, hence, different salary schedules.

P~ru~eteriz~tio~ of school district resources By parameterizing the amount of salary weight given to the factor associated with the

difficulty of the learning environment, in increments of $200, alternate “optimal” salary schedules can be derived. Table 4 summarizes the resulting alternate salary schedules. Alternate optimal salary schedules occurring as a result of parameterizing factor X3, supervisory responsibility in increments of $200 are recorded in Table 5. In Tables 6 and 7 the solutions obtained by the parameterization of salary levels are recorded. Table 6 lists the alternate salary schedules resulting from the parameterization, of beginning teacher aides salaries or the base salary in the district, in increments of $100.

114

JAMES EDWARD

BRUNO

TALILE4. bb?ZRNATEO~ALSALARYSCHEOULES WlTH FACTOR X,(LEARN1NG PARAMETERIZED IN INCREMENTS OF St00

Factors XI (parameterized) X? X, X, X, X, Xi X, X, District Cost (Objective Function) Function 1 (highest) (lowest) 2 3 4 5

250 527 100 100 50 1000 50 20 2402

450 327* 100 100 50 1000 50 24:;

ENVIRONMENT)

650 200 505 100 Z 50 20 2020

S11,916,248

Sl 1,940,038

S12,107.913

818,575 $14,860 915,642 S12,207 813,564 $8,429 $10,537 S7,102 s7.434 .S4,500

S18,575 $14,860 815,642 S12,207 $13.564 S8.429 510,537 S7.102 s7,434 %I,500

S19,198 $15,358 Sl6,167 812,277 s13.641 S8,488 SlO,610 $7,125 S7,484 S4.500

* Notice the $200 addition to salary factor X, is completely absorbed by salary factor X,, hence the salary schedule is identical to the original salary schedule in column 1. To prevent this from happening one merely places a lower bound of S527 for factor X, in the model.

TABLE 5. ALTERNATESALARYSCHEDULES

Factors XI X, Xs (parameterized) X, X, X.9 X; X, X, District Cost Function 1 (highest) (lowest) 2 (highest) (lowest) 3 (highest) (lowest) 4 (highest) (lowest) 5 (highest)

RESULTING FROM PARAMETERIZING FACTOR X,(EXPERIENCE) IN INCREMENTS OF s200

250 527 100 100 50 1000 50 20 2402

250 530 300 100 50 loo0 50 20 2199

250 534 500 100 50 1000 50 20 1996

S11,916,248 Sl8.575 514,860 $15,642 S 12,207 $13,564 58,429 810,537 $7,102 S7,434 S4,sOO

S11,918,625 Sl8.809 $15,047 515,839 $12,198 s13,553 $8,432 910,540 $7,099 s7,441 $4.500

$11,921,002 Sl9,042 S15.234 916.037 512,189 Sl3,542 S8.435 $10,544 S7,097 %7,448 S4,500

Linear Programming

Salary Evaluation in Negotiations

with Teacher Unions

115

TABLE 6. ALTERNATE SALARY SCHEDULES PARAMETERlZATlON OF SALARY LEVEL oj (BEGWNINGTEACHER AIDE)INI%REME;\TTsoF 8100

250 767 362 100 50 1000 50 20 2500 District Cost Function 1 (highest) (lowest) 2 (highest) (lowest) 3 (highest) (lowest) 4 (highest) (lowest) 5 (highest) Parameterized (lowest)

$13645,712 S21,289 617,161 ala,064 913,886 $15,429 $9,582 912,218 88,062 sa,525 $5,100

250 a08 421 100 50 1000 50 20 2500 Sl3.934.146

Sz,ia27 S17.559 518,483 Sl4,165 t15,739 S9,774 512,218 58,221 $8,696 S5,200

250 a48 481 100 50 1000 50 20 2500 Sl4,222,580 $22,366 517,958 Sla,904 %I4444 $16,049 $9,966 $12,458 88,381 58,876 $5,300

The parameterization of beginning teacher salaries (c,) in increments of $100 are recorded in the following table. TABLE 7. ALTERNATESALARYSCHEDULES PARAMETERIZATIONOF SALARY LEVEL O, (BEGINNINGTEACHER SALARY)IN INCREMENTS OF $200 Factor

Xl x2

x, x4 X6 & x7 X8 & Total District Cost Function 1 (highest) (lowest) 2 (highest) (lowest) 3 (highest) (lowest) 4 (highest Parameterized (lowest) 5 (highest) (lowest)

Determination

250 527 100 100 50 1000 50 20 2402

250 430 100 100 50 1000 50 20 2500

250 230 300 100 50 1000 50 20 2500

S11,916,248 SLa.575 $14,860 $15,642 S12,207 S13,564 58,429 $10,537 87,102 $7,434 $4,500

S11,916,930 518,440 s14.953 s 15,740 Sl2,500 si3,889 58,352 $10,440 S7,200 87,240 S4,500

S11,920,730 519,240 915,392 $16,140 S13,lOO 514,555 $8,733 NO,240 87,400 56,840 54,500

of individual salaries

To determine an individual salary, the school administrator merely multiplies the relative weight for the particular characteristic possessed by the individual times the factor weight calculated from the model.

116

JAMES EDWARD BRIJNO

Mathematically

this can be represented as

where : S = the individual’s salary xj = the relative weight of characteristicj found in factor i which is possessed by the individual Xi = the weight for factor i by solving the linear programming model.

To illustrate how the salary for school district personnel can be calculated, consider the salary schedule found in Table 3. A teacher (3X,) with a MA (4X4) have a classwide workload (2X0), in a difficult learning environment (3X,) with 2 years experience, with 7 additional college credits (1 X,) with 5 in-service credits (1 X,), with no distinctions (1 XJ and possessing no particular skill area (2X?) should receive as a salary. 3X,+2X,+3XJ+4X,+2X,+lX~+lX,+2X, 3(250) + 2(527.37) + 3(100) + 4(100) + 2(50) + 1000 + 50 + 20 + 4805.26 = $8481 If union demands were such, that the beginning teacher salary was S7400 and the salary schedule as found in Table 7 was adopted then this same teacher would receive 3X,+2X,+3X,+4X,+2X,+Xg+X,+X,+2Xg 3(250) + 2(230) + 3(300) + 4(100) + 2(50) + 1000 + 50 + 20 + 2(2500) =750+460+900+400+ 100+ 1000+50+20+5000 = $8680

CONCLUDING

REMARKS

The linear-programming approach to position/salary evaluation of school-district personnel allows a school district to calculate a system of relative weights, thereby establishing the relationship of one position to another, in quantitative terms. The approach proposed in this paper has several characteristics and advantages which distinguish it from previous salary-evaluation schemes in education : 1. It is an internally consistent evaluation scheme which is valid for all school-district functions considered in the model and which takes into consideration all the agreedupon factors which constitute those functions. 2. The model presents to the school district a more effective assessment of each individual’s relative worth to the school district, which can be reflected in terms of salary. 3. The model could be used to justify salary increases for school-district personnel and could play an important role in wage-salary negotiations with teacher unions and associations. 4. The model establishes a salary hierarchy consistent with the objectives of the school district but allows for highly qualified ‘personnel in one function to receive larger salaries than the lowest-qualified personnel in a higher function.

Linear Programming Salary Evaluation in Negotiations with Teacher Unions

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The model allows the school district to establish salary priorities (e.g., a school district can pay higher salaries to teachers in difficult teaching situations or to those in high-demand, low-supply teaching areas or, as demonstrated in this paper, beginning teacher salaries. The evaluation system encourages participation of teacher groups, administrators, PTA, and the school board in setting the objectives of the school, the functions or job descriptions of its personnel, the establishment of those factors necessary for performance of the particular function, and the rating of the characteristics which constitute each factor.

REFERENCES 1. CHARLESS. BENSON,T/E Economics ofPaNic Educutio~r, Houghton-Miff&z, Boston, 1965. 2. JOSEPHA. KERSHAWand ROLANDN. MCKEAN, Teacher Shortages and Salary Schedules,

The RAND Corporation, RM-3009-FF, February 1962. 3. ROBERTW. METZGER,Elementary Mathematical Programming, Wiley, New York, 1958. 4. FREDERICKP. REHMUSand HARVEY M. WAGNER, “Applying Linear Programming to Your Pay Structure,” Busin. Horiz., 4, 1963.