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Applying linear programming to your pay structure
FREDERICK P . R E H M U S AND HARVEY M . W A G N E R
APPLYING
LINEAR P R O G R A M M I N G
TO YOUR PAY S T R U C T U R E
ow much should a man be paid?" Many a manager has wished that he could use an objective and systematic approach to this sensitive question. Job evaluation has become accepted as an integral element of salary administration, since, despite its imperfections, it leads to more factual and rational compensation decisions. Specifically, it diagnoses a salary structure by determining the relative importance of the compensable elements of every job and assigns monetary equivalents to these ele-
H
Mr. Rehmus is Vice-President--Administratlon of the Midwestern Financial Corporation in Denver, and Mr. Wagner is Professor of Business Admlnls. ~ratlon at Stanford University.
WINTER, 1963
ments in order to bring about a consistent scale of compensation within a firm. It also aims at making the finn's scale generally competitive with those in the industry. As a consequence, the job evaluation approach discovers positions currently being incorrectly compensated and establishes a means whereby salaries for new positions are easily determined. Thus, when the results of a job evaluation study are fully implemented over a period of time, the compensation structure adjusts toward a system of adequate rewards and incentives for service. This article suggests how linear programming, a modern technique of management science usually applied to the production, distribution, marketing, and planning facets of a corporate enterprise, can be turned to-
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FREDERICK P. REHMUS AND HAnVEY M. WAGNER
90
ward perfecting a company's wage and salary program. Before getting into detail, two points must be made clear at the outset. First, we do not attempt to justify the job evaluation approach per se; to do so would require a discourse of at least equal length. 1 We realize, of course, that in bypassing a discussion of the pros and cons we forgo engagement in a number of pithy controversies. Our only contribution to these polemics is that our approach, insofar as it enhances the validity of job evaluation, lends tacit support to applying the method. This article also does not delve into the intricacies of obtaining the raw data to be used in a job evaluation program. The analysis begins just after the basic data have been collected. Second, we urge that our claim to a means of making job evaluation more objective and scientific be understood in the context of having to select one from among several possible approaches to wage and salary administration. One would be hard pressed to devise a method for properly setting compensation levels that did not in some way embrace human judgment. The approach to job evaluation suggested here eliminates the arbitrary elements usually present, reduces discretion to its essential level, and assists management to focus sharply on the multiple interacting influences that have to be accounted for in a compensation scheme. Thus, with this technique of management science, an executive needs to use his own judgment only where it is appropriate and necessary. The powerful scope of the method guarantees that both factual data and managerial goals are melded correctly. Perhaps the best way to state the contention that the approach is objective and scientific is to assert that
1 See Edwin F. Beal, "In Praise of Job Evaluation," California Management Review, V (Summer, ]963), 9-16 for a review of the approach's merits. Also see Arch Patton, Men, Money and Motivation (New York: McGraw-Hill Book Co., Inc., 1961) for a general picture of salary and wage administration practices.
with this method every salary analyst who faces the same raw data will arrive at the same compensation scale. This article tackles the problem of assigning proper weights to the compensable elements in a job evaluation system, a facet of the procedure that has often been subject to severe criticism. 2 A bias created in the selection of weights can invalidate sound analytical work done at earlier stages in the job evaluation process. Moreover, the possibility of misjudgment at this stage is great because an error here is less apparent, intuitively or empirically, than at the other stages. Hence, a technique that reduces or removes the element of judgment from factor weighting can substantially improve the process.
THE PROBLEM OF FACTOR WEIGHTING
The difficulties inherent is setting factor weights can be readily understood through a hypothetical example. In surveying an organization's job demands, a salary analyst has determined that three factors (A, B, and C) are the compensable elements of jobs in this organization. Salary will be based on the relative demands of individual jobs in terms of these three factors. Further, the analyst has observed six identifiable degrees for A, four for B, and five for C. These degrees are essentially rankings, arranged from the least to the greatest level of contribution. The complete factor evaluation structure appears in Table 1. In Table 2, five of the jobs analyzed are rated in terms of the evaluation factors. An examination of the above data indicates that the sum of the factor degree ratings does not correspond closely to the existing salaries for individual jobs. Two reasons may account for the discrepancies: ( 1 ) the sala-
2 A lucid critique of some of the factor-weighting techniques and a strong argument for an improved, more valid technique may be found in William M. Fox, "Purpose and Validity in Job Evaluation," Personnel 1ournal (October, 1962), pp. 432-37.
BUSINESS HORIZONS
APPLYING LINEAR PROGRAM2klING TO YOUR PAY STRUGTURE
T3xBLE 1
Factor Evaluation Structure: Factor Degree Levels
A1 A2 A3 A4 A5 A6
B1
C1
B2 B3 B4
C2 C3 C4 C5
TABLE 2
Rating of Five Jobs in Terms of Evaluation Factors Present Factor Sum of Faetor Monthly Degree Ratings Degree Ratings Salary
Job 1 Job 2 Job 3 Job 4 Job 5
A2 A4 A5 A5 A6
B4 B1 B2 B4 B4
C2 C4 C5 C4 C2
2+4+2= 8 4+1+4= 9 5+2+5=12 5+4+4=13 6-t-4+2=12
$ 800 1,000 1,100 1,400 1,500
ries for these jobs may be out of line at present, or (2) the relative importance of the factors and of the degree levels is not adequately represented by the numerical values given to factor degree ratings. In the design and implementation of actual job evaluation programs, these discrepancies are encountered almost universally. A high degree of management subjectivity usually enters into ferreting out the reasons for the discrepancies. Unfortunately, it is not possible with raw data such as those in Table 2 to answer separately the questions,"Are the data correct indicators of the compensable elements?" and "Which current salaries are unfair measures of contribution?" From the very start, one must recognize that the answers are inevitably mutually determined. This is not to say that in performing job evaluation management must ignore experience and common sense to become a slave to the data. For example, if a particular individual is paid an exceptional salary because of eireumstanees that are not reflected in the faetot ratings of his job, it would be patent nonsense for the analyst to use this individ-
WINTER, 1963
ual's salary in determining correct factor weights. On the contrary, management judgment must be applied not only in establishing job requirements but also in reviewing whether the current salaries significantly represent a historical condition that is no longer consistent with the job evaluation survey. We firmly believe that management can exercise this degree of judgment with far more objectivity than is possible in setting factor weights. Various techniques for weighting factors have been developed, and it is pre.cisely here that linear programming can provide improved, objectively based answers in contrast to the prevalent alternative approaches. TRADITIONAL TECHNIQUES F O R FACTOR WEIGHTING
The most common approach to solving the weighting problem is through the use of "best guess" weights for each factor. Let us assume that the existing salaries in Table 2 reflect a fair degree of rationality, except perhaps for some minor discrepancies that need adjusting. Then factor A might be given twice the weight of factors B and C, or some other system of weighting might be devised to bring the sum of the factor degrees more into line with salaries. Adding artificial factor degrees as needed in order to give logical salary relationships within the total evaluation structure offers further refinement. In the illustration in Table 2, a seventh degree level for factor A can be created. Then job 5 can be reevaluated to have a rating of A7 rather than A6. Consequently, the sum of the factor degrees for job 5. more closely con'esponds to the present salary. With this type of manipulation, factor degree A7 is artificially induced to make the evaluation structure more rational. In the hands of a skilled analyst, this type of adjustment technique can often produce satisfactory results. The obvious disadvantage of the approaeh is that it is wholly subjeetive and arbitrary. The appearance of the
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FREDERICK 1~. REHMUS AND HARVEY M. WAGNER
92
final structure might differ if another analyst performed the adjustments, and the approach is thus difficult to defend on any scientific basis. Standard multiple correlation and regression analysis is another technique for determining weights? Subjectivity is decreased to some extent with this method, and it is necessary only to express a general mathematical relation for the weights; the subsequent mathematical computations give the ultimate determination. A major disadvantage of this approach is that in actual practice the final result often proves to be unworkable because the calculated weights do not correlate closely enough with existing salaries; some weights are negative, and others seem to be disproportionately large or small. In an effort to correct these numerical inequities, the analyst once again must fall back on subjective adjustments. Other established techniques for solving the weighting problem are primarily variants of the two described above. A LINEARPROGRAMMINGAPPROACH With the development of linear programming techniques, an opportunity has been created to solve the weighting problem in job evaluation with minima/amounts of subjectivity and associated effort.4 The linear programming approach has as its initial goal the selection of the weights that explain most precisely the relationship between the factor
s An extensive discussion of this statistical technique for developing factor weights appears in Robert W. Gilmour, Industrial Wage and Salary Control (New York: John Wiley & Sons, Inc., 1956). 4An interesting pioneer application of linear programming was developed in A. Charnes, W. Cooper, and O. Ferguson, "Optimal Estimation of Executive Compensation by Linear Programming," Management Science, I (January, 1955), 138-51. Our experience with this approach has shown that it is not flexible enough in establishing factor weights to produce workable salary structures. The method proposed in the present article is different in that it provides freedom in selecting factor weights that is essential to the establishment of a rational compensation system.
ratings and existing salary. 5Stated in another fashion, it seeks a factor weighting system that minimizes the discrepancy between the salaries for given jobs and the factor ratings of these jobs. 6 With this system of weights, it is possible to examine the relationship of one job to another in quantitative terms and to establish logical salary differences between jobs in the organization. The only data required for analysis are the factor degree ratings and existing salary for each jobd To obtain the weights through linear programming, it is necessary to employ an electronic digital computer. Standard computer routines are readily available for solving linear programming problems. Job evaluation applications suggested in this article typically would require only a few minutes of computer time. It is important to note that a weighting structure developed through the use of linear programming should not (and ordinarily will not) completely "rationalize" the relationship between the weighted job ratings and existing salaries. Discrepancies will remain after the linear programming weights are used, and jobs where existing salaries are out of line may be identified through the presence of these discrepancies.
LINEAR PROGRAMMING APPLIED
Two sets of data are given below to demonstrate the effectiveness of the proposed linear programming method. The first set pertains to administrative and clerical posi-
5 For the mathematical description underlying this approach see H. M. Wagner, "Non-Linear Regression with Minimal Assumptions," Journal of the American Statistical Association, LVII (September, 1962), 572-78. 6 As previously, we assume for the discussion to follow that existing salaries are the most relevant of all available compensation data for determining factor weights and reiterate that this assumption must always come under management scrutiny. We return to this point at the end of the article. 7 In cases where an unusually large number of job classifications are being evaluated, a sample of jobs will give valid results.
BUSINESS HORIZONS
APPLYING LINEAR PROGRAM"hIING TO YOUI:( PAY STRUCTURE
TABLE 3
Base Data of Thirty-five Administrative and Clerical Job Classifications Job Factor Degree Ratings Factor D: Number of Knowledge, Factor A: Factor J: Job Average Positions Employees Experience Accuracy Judgment Demands Salary* Factor K:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30 31 32 33 34 35
17 2 33 3 6 7 9 4 32 1 11 1 2 14 2 2 2 1 24 1 12 1 8 14 12 3 2 1 7 2 10 2 5 1 16
1 1 3 3 3 2 4 3 4 3 3 4 5 6 7 7 7 7 7 7 8 9 10 10 9 10 9 10 9 10 10 12 11 12 14
1 1 2 2 3 3 2 3 2 3 2 3 3 3 4 4 4 4 4 5 6 5 6 7 6 5 5 7 6 7 6 7 7 8 9
1 1 1 1 2 1 2 2 2 1 1 2 2 3 4 3 4 3 4 5 5 6 7 8 7 5 5 8 5 8 6 7 6 8 10
6 3 4 4 4 4 4 4 4 5 5 4 4 2 1 2 1 2 3 1 3 1 3 1 1 6 1 1 2 3 1 3 2 1 3
.535 .623 .612 .635 .617 .662 .618 .602 .572 .758 .622 .660 .620 .667 .869 .767 .792 .833 .765 .925 .780 .860 .908 .915 .885 .822 .882 .857 .920 .918 .940 .980 1.000 .947 .947
*Each salary is expressed as a decimal fraction of the highest salary a m o n g the thirty-five job classifications.
tions, the second to executive positions. Equally satisfactory results are obtained from both applications.
based on thirty-five salaried job categories ranging from messenger to purchasing representative and systems analyst. Each job classification was rated in terms of four factors: Factor K: Knowledge and experience; 14 levels were identified ( K 1 . . . K14) Factor A: Accuracy; 9 degrees were identiffed ( A 1 . . . A9) Factor J: Judgment; 10 degrees were identiffed ( J 1 . . . J10) Factor D: Job demands; 6 levels were identified ( D 1 . . . D6). Table 3 shows the ratings for each of the thirty-five job categories, the number of employees in each job, and the average salary of employees in that classification, expressed as a fraction of the highest average salary (in this example, position 33). The conversion of average salaries to decimal fractions is numerically convenient, as will be seen below. Thus, if the highest average salary is $10,000, a decimal fraction of .660 corresponds to a salary of $6,600. It is a simple process to, relate to dollar salaries the resulting factor weights based on these decimal fractions.
Linear Program ]oh Equations The goal of a weighting system is to equate factor ratings with compensation. As noted previously, a weighting system cannot be expected to explain the salary structure completely, that is, to result in total evahmtions in which no deviations exist between the weights and current salaries. The linear program consequently contains a relation for each job that exhibits the possibility of a discrepancy occurring; in this example, thirtyfive job equations were included. Drawing from the data in Tab/e 3, the equations for job classifications 6, 16, and 26 are written as follows: Job 6 K2q-A3WJlq-D4-d6 =.662; Job 16 K7-I-A4-t-J3-}-D2-d16=.767; Job 26 K10q-A5q-JSq-D6 --- d26=.822.
AD~INISTlqATIVEANDCLElqICALPOSITIONS
Sample Data The data used in this example, taken from a major oil company, are
~V/NTE~R~ 1963
Notice that a discrepancy (d) between the total evaluation and an existing salary may be either positive or negative (-+), recog-
93
FREDERICK P. REHMUS AND HARVEY M. WAGNER
nizing that the job currently may be overpaid or underpaid.
Weighting System Obiective The linear programming technique finds a weighting system that minimizes the sum of the discrepancies for all the jobs being analyzed,s In some instances, it may be more appropriate to find weights that minimize the sum of the discrepancies among all individuals rather than among jobs. This objective is accomplished by weighting the deviations for individual job classifications by the number of individuals (N) in that job. 9
94
Constraints on the Weight Values In the selection of this firm's rating structure, which contains four factors and thirty-nine degrees, it was believed that each of these factors and each of the degrees within these factors should have some bearing on compensation. Past experience with workable job evaluation schemes indicated that no one factor should be permitted to dominate the evaluation structure. The need to give the linear program a frame of reference incorporating these considerations was satisfied by adding mild constraints on the permissible range of values for the optimal weights. 1° In this illustration, the following constraints were deemed appropriate: 1 No factor may contribute more than 35 per cent to the sum of all factor weights. This restriction is accomplished by limiting the weight of the top degree of each factor (K14, A9, J10, D6) to a value not exceeding .35. For example, the relation K14 _< .35 is added; a total of four such constr~fints are put into the program. 2 Each factor must have a positive weight in the final system. This is aecomplished in the illustration by limiting the value of the first degree of each factor (K1, A1, J1, D1) to a value of at least .05. For s T h a t is, m i n i m i z e ( d l -[- d2 -[- •. • q- clzs). s T h a t is, m i n i m i z e ( N l d l -[- N2d2 --[- Nad3 -~-
• . • q- N85das). lo The method of formulating appropriate constraints is described later.
TABLE4 Computer Factor Weights for Administrative and Clerical Evaluation Structure Factors Factor Degree
K
A
J
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14
.050 .057 .064 .071 .078 .120 .127 .134 .226 .235 .242 .249 .256 .263
.144 .224 .231 .308 .315 .322 .336 .343 .350
.050 .057 .064 .071 .078 .085 .092 .099 .106 .113
.245 .252 .259 .266 .284 .291
TABLE5 Computer Discrepancies Between Salaries and Evaluation Weights: Administrative and Clerical Positions Positions Discrepancy Positions 1 --19 2 .098 20 3 --21 4 .023 22 --4 2~ 5 .045 24 6 .006 25 7 (.015) 26 8 9 10 11 12 13 14 15 16 17 18
Discrepancy
-.160 -(.010) ----
(.081)
(.040)
27
.012
.131 -.043 ------.104 .002 .027 .068
28 29 30 31 32 33 34 35
(.058) .035 .003 .032 .033 .053
BUSINESS HORIZONS
APPLYING LINEAR PROGBAMIVIING TO YOUR PAY STRUCTURE
TABLE 6
Evaluation Structure for Executive Positions
Number of Factor
Factor Degrees
A-Responsibility for Personnel B-Responsibility for Revenue C-Responsibility for External Relations D-Responsibility for Physical Resources E-Planning, Problem Solving, and Creativeness F-Decision Making G-Knowledge, Skills, and Experience
18 14 15 14 10 11 10
TABLE 7
Base Data on Twenty-one Executive Positions
Positions
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Job Factor Degree Ratings A B C D E F G
Salary
18 17 16 15 9 12 14 15 6 13 10 11 8 10 5 4 3 7 1 5 2
1.000 .583 .716 .583 .333 .450 .375 .350 .333 .333 .375 .350 .333 .242 .308 .233 .225 .216 .250 .235 .180
WINTER, 1963
14 13 11 6 10 10 12 11 4 5 9 9 3 8 7 8 4 2 1 1 2
15 14 13 14 11 12 10 10 12 5 9 9 7 8 7 6 7 2 3 4 1
14 10 11 10 13 9 9 9 12 9 10 9 11 9 8 8 10 7 5 7 6 5 7 6 9 4 6 4 8 3 6 4 10 5 6 6 10 6 6 4 6 2 6 4 7 2 5 3 9 3 4 5 5 1 5 2 4 4 3 4 2 6 3 1 1 6 3 2 8 2 4 2 3 8 1 7 7 2 3 1 2 1 2 2
example, the relation K1 ~ .05 is added; four such constraints are in the program. 3 A difference must exist between the weights of successively higher degrees within a factor to ensure that each factor degree has some distinguishable bearing on the final results. In this study, the minimum acceptable difference between weights for successive factor degrees was judged to be .007. This is accomplished by adding a relation such as K2 => K1 q- .007; a total of thirty-five such constraints are used. The net result of these forty-three constraints is to allow a range of possible answers that avoided a swamping effect of any given factor.
Results of the L~nea~rProgram Table 4 illustrates the results of the program's application to the thirty-five clerical positions, showing weights assigned to each factor degree within the evaluation structure. The evaluation provides a minimum sum of discrepancies between the salaries paid and the factor ratings given. The magnitude of salary discrepancy for individual positions, found by applying these weights to the thirty-five positions, is identified in Table 5. A discrepancy reflects the differences between the sum of the factor weights for any position (K + A + J ÷ D), as shown in Table 4, and the average salary for that position, as shown in Table 3. It can be seen from Table 5 that salary reappraisal is required for the positions numbered 2, 10, 15, and 20, which are out of line in terms of the salaries paid. If the highest existing salary is $10,000, the largest discrepancy is $1,600 (position 20), indicating the advisability of reviewing this job and perhaps of establishing a range in which the maximum is lower than the average salary now being paid for it. EXECUTIVE POSITIONS
Sample Data Twenty-one executive positions in a public utility company provided
95
FREDERICKP. REHMUSAND Ha~avEy M. WACNER
the data used in the following example. The top position was that of the president and chief executive officer; the lowest paid position was that of the assistant treasurer. The evaluation structure included seven factors with varying numbers of degrees within factors, as shown in Table 6. Table 7 shows the ratings for each of the twentyone positions. Again, salaries have been expressed as a fraction of the top salary-in this case, the president's.
Program Formulation, Constraints, and Results As in the previous example, a
96
linear equation exists for each of the twentyone positions, and the program selects a set of weights that minimizes the sum of the possible discrepancies. The numerical values for the constraints placed on the executive position program differed from those used in the administrative and clerical program: 1 No factor at its highest degree level may have a value greater than .25. 2 No factor at its lowest degree may have a value less than .001. 3 The minimal difference between two successive factor levels must be .001. Table 8 shows the factor degree weights calculated by the program for the executive positions; Table 9 illustrates the discrepancies between weights and existing compensation levels, which identify positions in need of salary review (notably positions 2, 5, and 11). If the highest existing salary is $50,000, the largest discrepancy is $1,550 (position 5), indicating a current low level of compensation.
PREPARING FOR THE LINEAR PROGRAM DETERMINING SAMPLE SIZE
Satisfactory results were found in the preceding illustrations when only twenty-one and thirty-five jobs were to be evaluated by the program. It is not necessary to include every position in the scheme for determin-
ing factor weights, but more than twenty jobs should usually be included if a sample is used. Obviously, the degree of variance in the raw data will influence sample size and will be the major determinant of the number of jobs to be included. There are practical limitations on the extension of the sample and the number of positions that might be included. If the evaluation structure is too complex, the sheer size of the problem could defy present computer capability. In reaching a decision on sample size, we believe that an individual with a mathematical-statistical orientation should assist salary analysts. The approach requires the insights of both points of view. SETTING PROGRA/V[ CONSTRAINTS
Once the factors and factor levels to be included in an evaluation structure have been selected, each of them should have some bearing on the compensation decision ultimately reached. This requirement is carried into the formulation of the linear program by placing constraints on the permissible values of any factor degree. In the examples presented above, three kinds of constraints were used: (1) a maximum value on the weight of the highest rating of each factor; (2) a minimum value on the weight of the lowest rating of each; and (3) a minimum value on the difference between the weights of successive factor levels. In combination, these constraints required all factors and degrees to, have some distinguishable significance in the solution. For the two examples, we found it appropriate to use identical minimum and maximum constraining values for each factor, but in other applications experience and judgment may dictate the use of differing values. For example, it may be desirable to let one or two factors account for most of the weight. The maximum value for these factors would be accordingly large. The selection of values to be used in the constraints actually requires only a modi-
BUSINESS H O R I Z O N S
APPLYING LINEAR PItOGtlA~£MING T O YOUR P.4.Y $TRUCTLrltlg
TABLE 8
C o m p u t e r Factor Weights for Executive E v a l u a t i o n Structure Factor Degree A 1 .001 2 .002 3 .003 4 .004 5 .005 6 .006 7 .007 8 .014 9 .015 10 .016 11 .017 12 .018 13 .019 14 .020 15 .157 16 .158 17 .159 18 .250
B .001 .002 .003 .004 .005 .006 .007 .008 .025 .026 .027 .028 .029 .212
C .001 .002 .003 .026 .027 .028 .035 .036 .037 .038 .039 .040 .041 .042 .043
Factors D E .001 .001 .012 .014 .013 .015 .014 .016 .015 .017 .025 .018 .026 .019 .027 .020 .028 .021 .029 .022 .030 .031 .032 .033
F 225 .046 .047 .048 .049 .051 .138 .139 .140 .249 .250
G .114 .115 .180 .182 .183 .184 .185 .186 .187 .188
TABL~ 9 C o m p u t e r Discrepancies b e t w e e n Salaries and E v a l u a t i o n Weights: Executive Positions Positions 1 2 3 4 5 6 7 8 9 10 11
Discrepancy Positions 12 (.029) 13 ---14 -15 (.031) 16 17 -18 -19 20 21 .022 ----
-
Discrepancy
REFINING THE DATA
Obviously, the results of the suggested job evaluation p r o g r a m cannot b e any better than the r a w data used in the analysis. The linear p r o g r a m m i n g a p p r o a c h does not assist directly in the selection of factors and factor levels or in the establishment of ratings for particular jobs. But it can diagnose which factors are relatively most important, provided that the constraints on the weights are weak. T h e linear p r o g r a m m i n g approach should therefore b e viewed primarily as a means of effectively converting r a w evaluation data into c o m m e n s u r a t e compensation. T h r o u g h o u t this article, existing salaries h a v e b e e n used in the job equations of the linear program. I t is obvious that factor weights cannot b e evaluated in a vacuum; they m u s t be related to some kind of meaningful and systematic compensation data. O u r predilection for using existing salaries is b a s e d on the premise that they m a y well give the best indication of the relative value
-
205
cum of m a n a g e m e n t discretion. The values should allow sufficient flexibility so that the results are both meaningful a n d acceptable in terms of the general restrictions implicit in the rating structure selected. T h e values are a function of the n u m b e r of factors and
WINTER, 1963
factor levels in the job evaluation structure for which weights are being solved. 11 It is desirable to k e e p these constraints as mild as possible; the p r o g r a m is thus given a b r o a d r a n g e in which to seek an optimal solution.
u In the sample of administrative and clerical positions, the four factors seemed to be of equal importance, but the weighting system was kept flexible by permitting the maximum weight of any factor to be as large as .35. Similarly, in the sample of executive positions, the maximum allowable weight was set at .25, the smaller value resuiting from an increase in the number of factors to 7. The minimum allowable weight (.05 in the first example and .001 in the second example) reflected a reasonable minimum salary contribution for each of the factors. The difference between successive factor levels (.007 in the first example and .001 in the second) was determined by taking into account the smallest salary contribution that would reflect a distinguishable difference between factor levels, the range of salaries (from .535 to 1.000 in the first example, from .180 to 1.000 in the second) and the number of degree levels for each of the factors (from 6 to 14 levels in the first example, from 10 to 18 levels in the second).
97
FREDERICK P. REttMUS AND HAnV~ M. WAGNER
98
of jobs to an organization, and, consequently, the weights should be derived in terms of existing salary relationships. Although this approach would seem to treat somewhat lightly the inequities that may exist within an organization or in comparison to external salaries, two points must be made in defense of the method. First, employees are typically more concerned with internal than with external salary relationships. The employee is in a better position to evaluate his contribution and his salary relative to others in the same organization than to those with comparable positions in other organizations. Thus, internal salary relationships are of prime concern, and these are best expressed in terms of existing salaries. Second, the linear program by its very nature has the ability to compensate for internal inequities that may exist within an organization. Thus, ff several jobs are out of line in terms of internal relationships, they will not significantly distort the solution derived through linear programming. The design of the linear program incorporates an averaging effect that reduces the impact of individual jobs on the over-all solution. Despite the apparent usefulness of existing salaries, refining salary data in one of
several ways may nonetheless prove desirable in certain situations. As has been mentioned, gross inequities that are immediately apparent to the analyst through an examination of current salaries and position responsibilities should be excluded from the sample. If external relationships are important, industry data or salaries developed through special surveys can be used in place of actual salaries. We cannot stress too strongly, however, the need to ensure comparable job content if external salary information is used. Should any adjustments of the foregoing types be made, the linear programming approach suggested herein is applied with the salary information adjudged appropriate? 2 A manager must decide which data are relevant, no matter what approach he adopts. Linear programming assists him only in analyzing the full implications of these data. The desirability of the linear programming approach has been argued on the basis of its being objective and scientific. Management judgment and experience have been used where they are proper and direct, but arbitrary data manipulation has been discarded, and the results are reproducible by any analyst employing the same data. Although it is necessary to use a computer to perform the arithmetic calculations, the ready availability of linear programming routines and the negligible amount of computer time required make the approach practical for any firm desiring to apply it. As the two actual case examples have illustrated, the method of analysis is a powerful tool for establishing effective salary schedules.
12 In the event that management has, for one reason or another, established bench-mark salaries for specified jobs, these decisions cart easily be incorporated in the approach by eliminating the possibility of a dlserepancy in the associated positions. Other management policies can be incorporated, but doing so, of course, weakens the claim that the approach is an objective way to arrive at a rational salary structure.
BUSINESS HORIZONS
APPLYING
LINEAR P R O G R A M M I N G
TO YOUR PAY S T R U C T U R E
ow much should a man be paid?" Many a manager has wished that he could use an objective and systematic approach to this sensitive question. Job evaluation has become accepted as an integral element of salary administration, since, despite its imperfections, it leads to more factual and rational compensation decisions. Specifically, it diagnoses a salary structure by determining the relative importance of the compensable elements of every job and assigns monetary equivalents to these ele-
H
Mr. Rehmus is Vice-President--Administratlon of the Midwestern Financial Corporation in Denver, and Mr. Wagner is Professor of Business Admlnls. ~ratlon at Stanford University.
WINTER, 1963
ments in order to bring about a consistent scale of compensation within a firm. It also aims at making the finn's scale generally competitive with those in the industry. As a consequence, the job evaluation approach discovers positions currently being incorrectly compensated and establishes a means whereby salaries for new positions are easily determined. Thus, when the results of a job evaluation study are fully implemented over a period of time, the compensation structure adjusts toward a system of adequate rewards and incentives for service. This article suggests how linear programming, a modern technique of management science usually applied to the production, distribution, marketing, and planning facets of a corporate enterprise, can be turned to-
89
FREDERICK P. REHMUS AND HAnVEY M. WAGNER
90
ward perfecting a company's wage and salary program. Before getting into detail, two points must be made clear at the outset. First, we do not attempt to justify the job evaluation approach per se; to do so would require a discourse of at least equal length. 1 We realize, of course, that in bypassing a discussion of the pros and cons we forgo engagement in a number of pithy controversies. Our only contribution to these polemics is that our approach, insofar as it enhances the validity of job evaluation, lends tacit support to applying the method. This article also does not delve into the intricacies of obtaining the raw data to be used in a job evaluation program. The analysis begins just after the basic data have been collected. Second, we urge that our claim to a means of making job evaluation more objective and scientific be understood in the context of having to select one from among several possible approaches to wage and salary administration. One would be hard pressed to devise a method for properly setting compensation levels that did not in some way embrace human judgment. The approach to job evaluation suggested here eliminates the arbitrary elements usually present, reduces discretion to its essential level, and assists management to focus sharply on the multiple interacting influences that have to be accounted for in a compensation scheme. Thus, with this technique of management science, an executive needs to use his own judgment only where it is appropriate and necessary. The powerful scope of the method guarantees that both factual data and managerial goals are melded correctly. Perhaps the best way to state the contention that the approach is objective and scientific is to assert that
1 See Edwin F. Beal, "In Praise of Job Evaluation," California Management Review, V (Summer, ]963), 9-16 for a review of the approach's merits. Also see Arch Patton, Men, Money and Motivation (New York: McGraw-Hill Book Co., Inc., 1961) for a general picture of salary and wage administration practices.
with this method every salary analyst who faces the same raw data will arrive at the same compensation scale. This article tackles the problem of assigning proper weights to the compensable elements in a job evaluation system, a facet of the procedure that has often been subject to severe criticism. 2 A bias created in the selection of weights can invalidate sound analytical work done at earlier stages in the job evaluation process. Moreover, the possibility of misjudgment at this stage is great because an error here is less apparent, intuitively or empirically, than at the other stages. Hence, a technique that reduces or removes the element of judgment from factor weighting can substantially improve the process.
THE PROBLEM OF FACTOR WEIGHTING
The difficulties inherent is setting factor weights can be readily understood through a hypothetical example. In surveying an organization's job demands, a salary analyst has determined that three factors (A, B, and C) are the compensable elements of jobs in this organization. Salary will be based on the relative demands of individual jobs in terms of these three factors. Further, the analyst has observed six identifiable degrees for A, four for B, and five for C. These degrees are essentially rankings, arranged from the least to the greatest level of contribution. The complete factor evaluation structure appears in Table 1. In Table 2, five of the jobs analyzed are rated in terms of the evaluation factors. An examination of the above data indicates that the sum of the factor degree ratings does not correspond closely to the existing salaries for individual jobs. Two reasons may account for the discrepancies: ( 1 ) the sala-
2 A lucid critique of some of the factor-weighting techniques and a strong argument for an improved, more valid technique may be found in William M. Fox, "Purpose and Validity in Job Evaluation," Personnel 1ournal (October, 1962), pp. 432-37.
BUSINESS HORIZONS
APPLYING LINEAR PROGRAM2klING TO YOUR PAY STRUGTURE
T3xBLE 1
Factor Evaluation Structure: Factor Degree Levels
A1 A2 A3 A4 A5 A6
B1
C1
B2 B3 B4
C2 C3 C4 C5
TABLE 2
Rating of Five Jobs in Terms of Evaluation Factors Present Factor Sum of Faetor Monthly Degree Ratings Degree Ratings Salary
Job 1 Job 2 Job 3 Job 4 Job 5
A2 A4 A5 A5 A6
B4 B1 B2 B4 B4
C2 C4 C5 C4 C2
2+4+2= 8 4+1+4= 9 5+2+5=12 5+4+4=13 6-t-4+2=12
$ 800 1,000 1,100 1,400 1,500
ries for these jobs may be out of line at present, or (2) the relative importance of the factors and of the degree levels is not adequately represented by the numerical values given to factor degree ratings. In the design and implementation of actual job evaluation programs, these discrepancies are encountered almost universally. A high degree of management subjectivity usually enters into ferreting out the reasons for the discrepancies. Unfortunately, it is not possible with raw data such as those in Table 2 to answer separately the questions,"Are the data correct indicators of the compensable elements?" and "Which current salaries are unfair measures of contribution?" From the very start, one must recognize that the answers are inevitably mutually determined. This is not to say that in performing job evaluation management must ignore experience and common sense to become a slave to the data. For example, if a particular individual is paid an exceptional salary because of eireumstanees that are not reflected in the faetot ratings of his job, it would be patent nonsense for the analyst to use this individ-
WINTER, 1963
ual's salary in determining correct factor weights. On the contrary, management judgment must be applied not only in establishing job requirements but also in reviewing whether the current salaries significantly represent a historical condition that is no longer consistent with the job evaluation survey. We firmly believe that management can exercise this degree of judgment with far more objectivity than is possible in setting factor weights. Various techniques for weighting factors have been developed, and it is pre.cisely here that linear programming can provide improved, objectively based answers in contrast to the prevalent alternative approaches. TRADITIONAL TECHNIQUES F O R FACTOR WEIGHTING
The most common approach to solving the weighting problem is through the use of "best guess" weights for each factor. Let us assume that the existing salaries in Table 2 reflect a fair degree of rationality, except perhaps for some minor discrepancies that need adjusting. Then factor A might be given twice the weight of factors B and C, or some other system of weighting might be devised to bring the sum of the factor degrees more into line with salaries. Adding artificial factor degrees as needed in order to give logical salary relationships within the total evaluation structure offers further refinement. In the illustration in Table 2, a seventh degree level for factor A can be created. Then job 5 can be reevaluated to have a rating of A7 rather than A6. Consequently, the sum of the factor degrees for job 5. more closely con'esponds to the present salary. With this type of manipulation, factor degree A7 is artificially induced to make the evaluation structure more rational. In the hands of a skilled analyst, this type of adjustment technique can often produce satisfactory results. The obvious disadvantage of the approaeh is that it is wholly subjeetive and arbitrary. The appearance of the
91
FREDERICK 1~. REHMUS AND HARVEY M. WAGNER
92
final structure might differ if another analyst performed the adjustments, and the approach is thus difficult to defend on any scientific basis. Standard multiple correlation and regression analysis is another technique for determining weights? Subjectivity is decreased to some extent with this method, and it is necessary only to express a general mathematical relation for the weights; the subsequent mathematical computations give the ultimate determination. A major disadvantage of this approach is that in actual practice the final result often proves to be unworkable because the calculated weights do not correlate closely enough with existing salaries; some weights are negative, and others seem to be disproportionately large or small. In an effort to correct these numerical inequities, the analyst once again must fall back on subjective adjustments. Other established techniques for solving the weighting problem are primarily variants of the two described above. A LINEARPROGRAMMINGAPPROACH With the development of linear programming techniques, an opportunity has been created to solve the weighting problem in job evaluation with minima/amounts of subjectivity and associated effort.4 The linear programming approach has as its initial goal the selection of the weights that explain most precisely the relationship between the factor
s An extensive discussion of this statistical technique for developing factor weights appears in Robert W. Gilmour, Industrial Wage and Salary Control (New York: John Wiley & Sons, Inc., 1956). 4An interesting pioneer application of linear programming was developed in A. Charnes, W. Cooper, and O. Ferguson, "Optimal Estimation of Executive Compensation by Linear Programming," Management Science, I (January, 1955), 138-51. Our experience with this approach has shown that it is not flexible enough in establishing factor weights to produce workable salary structures. The method proposed in the present article is different in that it provides freedom in selecting factor weights that is essential to the establishment of a rational compensation system.
ratings and existing salary. 5Stated in another fashion, it seeks a factor weighting system that minimizes the discrepancy between the salaries for given jobs and the factor ratings of these jobs. 6 With this system of weights, it is possible to examine the relationship of one job to another in quantitative terms and to establish logical salary differences between jobs in the organization. The only data required for analysis are the factor degree ratings and existing salary for each jobd To obtain the weights through linear programming, it is necessary to employ an electronic digital computer. Standard computer routines are readily available for solving linear programming problems. Job evaluation applications suggested in this article typically would require only a few minutes of computer time. It is important to note that a weighting structure developed through the use of linear programming should not (and ordinarily will not) completely "rationalize" the relationship between the weighted job ratings and existing salaries. Discrepancies will remain after the linear programming weights are used, and jobs where existing salaries are out of line may be identified through the presence of these discrepancies.
LINEAR PROGRAMMING APPLIED
Two sets of data are given below to demonstrate the effectiveness of the proposed linear programming method. The first set pertains to administrative and clerical posi-
5 For the mathematical description underlying this approach see H. M. Wagner, "Non-Linear Regression with Minimal Assumptions," Journal of the American Statistical Association, LVII (September, 1962), 572-78. 6 As previously, we assume for the discussion to follow that existing salaries are the most relevant of all available compensation data for determining factor weights and reiterate that this assumption must always come under management scrutiny. We return to this point at the end of the article. 7 In cases where an unusually large number of job classifications are being evaluated, a sample of jobs will give valid results.
BUSINESS HORIZONS
APPLYING LINEAR PROGRAM"hIING TO YOUI:( PAY STRUCTURE
TABLE 3
Base Data of Thirty-five Administrative and Clerical Job Classifications Job Factor Degree Ratings Factor D: Number of Knowledge, Factor A: Factor J: Job Average Positions Employees Experience Accuracy Judgment Demands Salary* Factor K:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30 31 32 33 34 35
17 2 33 3 6 7 9 4 32 1 11 1 2 14 2 2 2 1 24 1 12 1 8 14 12 3 2 1 7 2 10 2 5 1 16
1 1 3 3 3 2 4 3 4 3 3 4 5 6 7 7 7 7 7 7 8 9 10 10 9 10 9 10 9 10 10 12 11 12 14
1 1 2 2 3 3 2 3 2 3 2 3 3 3 4 4 4 4 4 5 6 5 6 7 6 5 5 7 6 7 6 7 7 8 9
1 1 1 1 2 1 2 2 2 1 1 2 2 3 4 3 4 3 4 5 5 6 7 8 7 5 5 8 5 8 6 7 6 8 10
6 3 4 4 4 4 4 4 4 5 5 4 4 2 1 2 1 2 3 1 3 1 3 1 1 6 1 1 2 3 1 3 2 1 3
.535 .623 .612 .635 .617 .662 .618 .602 .572 .758 .622 .660 .620 .667 .869 .767 .792 .833 .765 .925 .780 .860 .908 .915 .885 .822 .882 .857 .920 .918 .940 .980 1.000 .947 .947
*Each salary is expressed as a decimal fraction of the highest salary a m o n g the thirty-five job classifications.
tions, the second to executive positions. Equally satisfactory results are obtained from both applications.
based on thirty-five salaried job categories ranging from messenger to purchasing representative and systems analyst. Each job classification was rated in terms of four factors: Factor K: Knowledge and experience; 14 levels were identified ( K 1 . . . K14) Factor A: Accuracy; 9 degrees were identiffed ( A 1 . . . A9) Factor J: Judgment; 10 degrees were identiffed ( J 1 . . . J10) Factor D: Job demands; 6 levels were identified ( D 1 . . . D6). Table 3 shows the ratings for each of the thirty-five job categories, the number of employees in each job, and the average salary of employees in that classification, expressed as a fraction of the highest average salary (in this example, position 33). The conversion of average salaries to decimal fractions is numerically convenient, as will be seen below. Thus, if the highest average salary is $10,000, a decimal fraction of .660 corresponds to a salary of $6,600. It is a simple process to, relate to dollar salaries the resulting factor weights based on these decimal fractions.
Linear Program ]oh Equations The goal of a weighting system is to equate factor ratings with compensation. As noted previously, a weighting system cannot be expected to explain the salary structure completely, that is, to result in total evahmtions in which no deviations exist between the weights and current salaries. The linear program consequently contains a relation for each job that exhibits the possibility of a discrepancy occurring; in this example, thirtyfive job equations were included. Drawing from the data in Tab/e 3, the equations for job classifications 6, 16, and 26 are written as follows: Job 6 K2q-A3WJlq-D4-d6 =.662; Job 16 K7-I-A4-t-J3-}-D2-d16=.767; Job 26 K10q-A5q-JSq-D6 --- d26=.822.
AD~INISTlqATIVEANDCLElqICALPOSITIONS
Sample Data The data used in this example, taken from a major oil company, are
~V/NTE~R~ 1963
Notice that a discrepancy (d) between the total evaluation and an existing salary may be either positive or negative (-+), recog-
93
FREDERICK P. REHMUS AND HARVEY M. WAGNER
nizing that the job currently may be overpaid or underpaid.
Weighting System Obiective The linear programming technique finds a weighting system that minimizes the sum of the discrepancies for all the jobs being analyzed,s In some instances, it may be more appropriate to find weights that minimize the sum of the discrepancies among all individuals rather than among jobs. This objective is accomplished by weighting the deviations for individual job classifications by the number of individuals (N) in that job. 9
94
Constraints on the Weight Values In the selection of this firm's rating structure, which contains four factors and thirty-nine degrees, it was believed that each of these factors and each of the degrees within these factors should have some bearing on compensation. Past experience with workable job evaluation schemes indicated that no one factor should be permitted to dominate the evaluation structure. The need to give the linear program a frame of reference incorporating these considerations was satisfied by adding mild constraints on the permissible range of values for the optimal weights. 1° In this illustration, the following constraints were deemed appropriate: 1 No factor may contribute more than 35 per cent to the sum of all factor weights. This restriction is accomplished by limiting the weight of the top degree of each factor (K14, A9, J10, D6) to a value not exceeding .35. For example, the relation K14 _< .35 is added; a total of four such constr~fints are put into the program. 2 Each factor must have a positive weight in the final system. This is aecomplished in the illustration by limiting the value of the first degree of each factor (K1, A1, J1, D1) to a value of at least .05. For s T h a t is, m i n i m i z e ( d l -[- d2 -[- •. • q- clzs). s T h a t is, m i n i m i z e ( N l d l -[- N2d2 --[- Nad3 -~-
• . • q- N85das). lo The method of formulating appropriate constraints is described later.
TABLE4 Computer Factor Weights for Administrative and Clerical Evaluation Structure Factors Factor Degree
K
A
J
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14
.050 .057 .064 .071 .078 .120 .127 .134 .226 .235 .242 .249 .256 .263
.144 .224 .231 .308 .315 .322 .336 .343 .350
.050 .057 .064 .071 .078 .085 .092 .099 .106 .113
.245 .252 .259 .266 .284 .291
TABLE5 Computer Discrepancies Between Salaries and Evaluation Weights: Administrative and Clerical Positions Positions Discrepancy Positions 1 --19 2 .098 20 3 --21 4 .023 22 --4 2~ 5 .045 24 6 .006 25 7 (.015) 26 8 9 10 11 12 13 14 15 16 17 18
Discrepancy
-.160 -(.010) ----
(.081)
(.040)
27
.012
.131 -.043 ------.104 .002 .027 .068
28 29 30 31 32 33 34 35
(.058) .035 .003 .032 .033 .053
BUSINESS HORIZONS
APPLYING LINEAR PROGBAMIVIING TO YOUR PAY STRUCTURE
TABLE 6
Evaluation Structure for Executive Positions
Number of Factor
Factor Degrees
A-Responsibility for Personnel B-Responsibility for Revenue C-Responsibility for External Relations D-Responsibility for Physical Resources E-Planning, Problem Solving, and Creativeness F-Decision Making G-Knowledge, Skills, and Experience
18 14 15 14 10 11 10
TABLE 7
Base Data on Twenty-one Executive Positions
Positions
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Job Factor Degree Ratings A B C D E F G
Salary
18 17 16 15 9 12 14 15 6 13 10 11 8 10 5 4 3 7 1 5 2
1.000 .583 .716 .583 .333 .450 .375 .350 .333 .333 .375 .350 .333 .242 .308 .233 .225 .216 .250 .235 .180
WINTER, 1963
14 13 11 6 10 10 12 11 4 5 9 9 3 8 7 8 4 2 1 1 2
15 14 13 14 11 12 10 10 12 5 9 9 7 8 7 6 7 2 3 4 1
14 10 11 10 13 9 9 9 12 9 10 9 11 9 8 8 10 7 5 7 6 5 7 6 9 4 6 4 8 3 6 4 10 5 6 6 10 6 6 4 6 2 6 4 7 2 5 3 9 3 4 5 5 1 5 2 4 4 3 4 2 6 3 1 1 6 3 2 8 2 4 2 3 8 1 7 7 2 3 1 2 1 2 2
example, the relation K1 ~ .05 is added; four such constraints are in the program. 3 A difference must exist between the weights of successively higher degrees within a factor to ensure that each factor degree has some distinguishable bearing on the final results. In this study, the minimum acceptable difference between weights for successive factor degrees was judged to be .007. This is accomplished by adding a relation such as K2 => K1 q- .007; a total of thirty-five such constraints are used. The net result of these forty-three constraints is to allow a range of possible answers that avoided a swamping effect of any given factor.
Results of the L~nea~rProgram Table 4 illustrates the results of the program's application to the thirty-five clerical positions, showing weights assigned to each factor degree within the evaluation structure. The evaluation provides a minimum sum of discrepancies between the salaries paid and the factor ratings given. The magnitude of salary discrepancy for individual positions, found by applying these weights to the thirty-five positions, is identified in Table 5. A discrepancy reflects the differences between the sum of the factor weights for any position (K + A + J ÷ D), as shown in Table 4, and the average salary for that position, as shown in Table 3. It can be seen from Table 5 that salary reappraisal is required for the positions numbered 2, 10, 15, and 20, which are out of line in terms of the salaries paid. If the highest existing salary is $10,000, the largest discrepancy is $1,600 (position 20), indicating the advisability of reviewing this job and perhaps of establishing a range in which the maximum is lower than the average salary now being paid for it. EXECUTIVE POSITIONS
Sample Data Twenty-one executive positions in a public utility company provided
95
FREDERICKP. REHMUSAND Ha~avEy M. WACNER
the data used in the following example. The top position was that of the president and chief executive officer; the lowest paid position was that of the assistant treasurer. The evaluation structure included seven factors with varying numbers of degrees within factors, as shown in Table 6. Table 7 shows the ratings for each of the twentyone positions. Again, salaries have been expressed as a fraction of the top salary-in this case, the president's.
Program Formulation, Constraints, and Results As in the previous example, a
96
linear equation exists for each of the twentyone positions, and the program selects a set of weights that minimizes the sum of the possible discrepancies. The numerical values for the constraints placed on the executive position program differed from those used in the administrative and clerical program: 1 No factor at its highest degree level may have a value greater than .25. 2 No factor at its lowest degree may have a value less than .001. 3 The minimal difference between two successive factor levels must be .001. Table 8 shows the factor degree weights calculated by the program for the executive positions; Table 9 illustrates the discrepancies between weights and existing compensation levels, which identify positions in need of salary review (notably positions 2, 5, and 11). If the highest existing salary is $50,000, the largest discrepancy is $1,550 (position 5), indicating a current low level of compensation.
PREPARING FOR THE LINEAR PROGRAM DETERMINING SAMPLE SIZE
Satisfactory results were found in the preceding illustrations when only twenty-one and thirty-five jobs were to be evaluated by the program. It is not necessary to include every position in the scheme for determin-
ing factor weights, but more than twenty jobs should usually be included if a sample is used. Obviously, the degree of variance in the raw data will influence sample size and will be the major determinant of the number of jobs to be included. There are practical limitations on the extension of the sample and the number of positions that might be included. If the evaluation structure is too complex, the sheer size of the problem could defy present computer capability. In reaching a decision on sample size, we believe that an individual with a mathematical-statistical orientation should assist salary analysts. The approach requires the insights of both points of view. SETTING PROGRA/V[ CONSTRAINTS
Once the factors and factor levels to be included in an evaluation structure have been selected, each of them should have some bearing on the compensation decision ultimately reached. This requirement is carried into the formulation of the linear program by placing constraints on the permissible values of any factor degree. In the examples presented above, three kinds of constraints were used: (1) a maximum value on the weight of the highest rating of each factor; (2) a minimum value on the weight of the lowest rating of each; and (3) a minimum value on the difference between the weights of successive factor levels. In combination, these constraints required all factors and degrees to, have some distinguishable significance in the solution. For the two examples, we found it appropriate to use identical minimum and maximum constraining values for each factor, but in other applications experience and judgment may dictate the use of differing values. For example, it may be desirable to let one or two factors account for most of the weight. The maximum value for these factors would be accordingly large. The selection of values to be used in the constraints actually requires only a modi-
BUSINESS H O R I Z O N S
APPLYING LINEAR PItOGtlA~£MING T O YOUR P.4.Y $TRUCTLrltlg
TABLE 8
C o m p u t e r Factor Weights for Executive E v a l u a t i o n Structure Factor Degree A 1 .001 2 .002 3 .003 4 .004 5 .005 6 .006 7 .007 8 .014 9 .015 10 .016 11 .017 12 .018 13 .019 14 .020 15 .157 16 .158 17 .159 18 .250
B .001 .002 .003 .004 .005 .006 .007 .008 .025 .026 .027 .028 .029 .212
C .001 .002 .003 .026 .027 .028 .035 .036 .037 .038 .039 .040 .041 .042 .043
Factors D E .001 .001 .012 .014 .013 .015 .014 .016 .015 .017 .025 .018 .026 .019 .027 .020 .028 .021 .029 .022 .030 .031 .032 .033
F 225 .046 .047 .048 .049 .051 .138 .139 .140 .249 .250
G .114 .115 .180 .182 .183 .184 .185 .186 .187 .188
TABL~ 9 C o m p u t e r Discrepancies b e t w e e n Salaries and E v a l u a t i o n Weights: Executive Positions Positions 1 2 3 4 5 6 7 8 9 10 11
Discrepancy Positions 12 (.029) 13 ---14 -15 (.031) 16 17 -18 -19 20 21 .022 ----
-
Discrepancy
REFINING THE DATA
Obviously, the results of the suggested job evaluation p r o g r a m cannot b e any better than the r a w data used in the analysis. The linear p r o g r a m m i n g a p p r o a c h does not assist directly in the selection of factors and factor levels or in the establishment of ratings for particular jobs. But it can diagnose which factors are relatively most important, provided that the constraints on the weights are weak. T h e linear p r o g r a m m i n g approach should therefore b e viewed primarily as a means of effectively converting r a w evaluation data into c o m m e n s u r a t e compensation. T h r o u g h o u t this article, existing salaries h a v e b e e n used in the job equations of the linear program. I t is obvious that factor weights cannot b e evaluated in a vacuum; they m u s t be related to some kind of meaningful and systematic compensation data. O u r predilection for using existing salaries is b a s e d on the premise that they m a y well give the best indication of the relative value
-
205
cum of m a n a g e m e n t discretion. The values should allow sufficient flexibility so that the results are both meaningful a n d acceptable in terms of the general restrictions implicit in the rating structure selected. T h e values are a function of the n u m b e r of factors and
WINTER, 1963
factor levels in the job evaluation structure for which weights are being solved. 11 It is desirable to k e e p these constraints as mild as possible; the p r o g r a m is thus given a b r o a d r a n g e in which to seek an optimal solution.
u In the sample of administrative and clerical positions, the four factors seemed to be of equal importance, but the weighting system was kept flexible by permitting the maximum weight of any factor to be as large as .35. Similarly, in the sample of executive positions, the maximum allowable weight was set at .25, the smaller value resuiting from an increase in the number of factors to 7. The minimum allowable weight (.05 in the first example and .001 in the second example) reflected a reasonable minimum salary contribution for each of the factors. The difference between successive factor levels (.007 in the first example and .001 in the second) was determined by taking into account the smallest salary contribution that would reflect a distinguishable difference between factor levels, the range of salaries (from .535 to 1.000 in the first example, from .180 to 1.000 in the second) and the number of degree levels for each of the factors (from 6 to 14 levels in the first example, from 10 to 18 levels in the second).
97
FREDERICK P. REttMUS AND HAnV~ M. WAGNER
98
of jobs to an organization, and, consequently, the weights should be derived in terms of existing salary relationships. Although this approach would seem to treat somewhat lightly the inequities that may exist within an organization or in comparison to external salaries, two points must be made in defense of the method. First, employees are typically more concerned with internal than with external salary relationships. The employee is in a better position to evaluate his contribution and his salary relative to others in the same organization than to those with comparable positions in other organizations. Thus, internal salary relationships are of prime concern, and these are best expressed in terms of existing salaries. Second, the linear program by its very nature has the ability to compensate for internal inequities that may exist within an organization. Thus, ff several jobs are out of line in terms of internal relationships, they will not significantly distort the solution derived through linear programming. The design of the linear program incorporates an averaging effect that reduces the impact of individual jobs on the over-all solution. Despite the apparent usefulness of existing salaries, refining salary data in one of
several ways may nonetheless prove desirable in certain situations. As has been mentioned, gross inequities that are immediately apparent to the analyst through an examination of current salaries and position responsibilities should be excluded from the sample. If external relationships are important, industry data or salaries developed through special surveys can be used in place of actual salaries. We cannot stress too strongly, however, the need to ensure comparable job content if external salary information is used. Should any adjustments of the foregoing types be made, the linear programming approach suggested herein is applied with the salary information adjudged appropriate? 2 A manager must decide which data are relevant, no matter what approach he adopts. Linear programming assists him only in analyzing the full implications of these data. The desirability of the linear programming approach has been argued on the basis of its being objective and scientific. Management judgment and experience have been used where they are proper and direct, but arbitrary data manipulation has been discarded, and the results are reproducible by any analyst employing the same data. Although it is necessary to use a computer to perform the arithmetic calculations, the ready availability of linear programming routines and the negligible amount of computer time required make the approach practical for any firm desiring to apply it. As the two actual case examples have illustrated, the method of analysis is a powerful tool for establishing effective salary schedules.
12 In the event that management has, for one reason or another, established bench-mark salaries for specified jobs, these decisions cart easily be incorporated in the approach by eliminating the possibility of a dlserepancy in the associated positions. Other management policies can be incorporated, but doing so, of course, weakens the claim that the approach is an objective way to arrive at a rational salary structure.
BUSINESS HORIZONS