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Mathematical programming models to determine civil service salaries
Mathematical programming models to determine civil service salaries servants with the compensation system presently employed. The relative worth of an employee is determined by a fixed step schedule I. The two charac. teristics of an individual employee included in this schedule are civil service grade and the number of years of experience in the particular grade. The rigidity of the traditional fixed step salary schedule has resulted in problems for civil servants and government agencies. Specifically, since the salary scheme does not consider other factors such as special licences, additional job.related education and responsibilities of the position, to name but a few, many civil servants do not feel that they are being fairly and adequately compensated when compared to other civil service employees with the same Civil Service Rank. Moreover, the salary structure does not reflect managements' priorities and resources. This would enable management to offer added incentive to entice employees to fill or remain in "lackluster" but critical positions. This feature is conspicuously absent fiom the fixed step salary scheme and today is causing problems fur public service departments and agencies. For example, the New York State Department of Transportation has recently put greater emphasis on in-house design calling for a shift of personnel from the Construction Section to tl, e Design Section. (Prior to austerity much of this design work was performed by consultants.) An assignment to construction gives the engineer an opportunity to work outdoors and a feeling of accomplishment, as he can readily see the work completed, among other benefits. Design, on the other hand, presents the traditional office setting with all its constraints. Under the present salary schedule, management can do nothing (financially) for the engineer to entice him to come or even remain in the Design Section.
F r a n k J. F A B O Z Z I , School of Business, Hofs~-a Un~ersity, Hempstead, N Y 11550, U.S.A. A l f r e d W. B A C H N ~ ! R New York State Departm~.nt of Transportation, New York, NY, U.S.A. Received 19 October 1977 Revised 3 April 1978 Increased criticism has been directed in recent years at the present fixed .qtep salary method used for compensating civil servants. Under this method the relative monetary value of an individual t~ he department or agency is bated only on two factors: civil service grade and the number of years of experience in the particular grade. This paper describes how two mathematical programming methods, linear pro~amming and goal programming, can be applied to determine a civil service salary structure for the New York State Department of Transportation (Region 10). The mathematical salary model employed was developed by J.E. Bruno. I, Introduction In the civil service the salary compensation of each employee is open, and it is therefore of primary importance that each employee feels that he or she is being not only adequately compensated, but also fairly compensated when compared to fellow employees, their duties, responsibilities, and expertise. We wish to thank the two referees of this journal, Clive Purkiss and Kees Verhoeven, and Lesley Browder for several helpful criticisms and suggestions. John Pizzariella provided valuab~_eprogramming assistance. Of course, the ~xaditional caveat concerning .~ny remaining errors applies. The views expressed in this paper represent those of the author. Salaries are a delicate matter, even among mature adults. They satisfy ego and gwe the worker prestige besides satisfying economic needs. A poorly designed (open or secret) salary eomper, sation system can only create misunderstanding, frustration and low morale. There is a great deal of discontent among civil
1 Historically, the fixed step salary schedule was implemented to guarantee civil servants, who were prohibited from joining collective bargaining units (unions), future salary raises [8, p.40]. When legislation mandated the civil servant to be represented by a bargaining unit, the fixed step salary schedule was a retained vestige; perhaps this was due to employee lack of confidence in the early bargaining units.
© North-Holland Publishing Company European Journai of Operational Research 3 (!979) i 90-198. 190
F.Z Fabozzi, A. W. Bachner / MP models to determine civil service ~alaries
The "merit system" employed in the private sector has been proposed as an alternative compensation system for the public sector. This system ~rovides for salary movements based on a set of objectives established and reviewed by management. The "merit system" is strongly supported by Administration and opposed by the Civil Service Employees Association (CSEA) [51. The "merit system" requires the development of objectives and the necessary criteria for evaluation. Herein lies the difficulty. There are presently no acceptable methods for measuring the effectiveness of many civil service employees. The evaluation then becomes qualitative and highly subjective leaving no recourse for debate. It is this qualitative and subjective evaluation that repulses the CSEA. In addition, the inability of the employee to duplicate the administration's evaluation compounds the employee's sense of frustration and defeat [7], and this detriment disqualifies the "merit system" as a plausible salary scheme by itself. However, it can be incorporated as one factor in the salary evaluation model described in this paper. The purpose of this paper is to demonstrate how mathematical programming can be used to develop a canpensation system which takes into consideration several characteristics of a job function and also reflects the objectives of the department or agency. The model will be applied to the New York State Department of Transportation, Region 10, where the traditional fixed step salary schedule is tl:e present method of salary evaluation for engineering personnel. Clmrnes et al. [2] and Relunus and Wagner [6] have suggested that linear programming could be used to determine executive compensation. Recently, Bruno [ 1] ires developed a linear programming salary evaluation model to compensate school district personnel who are presently compensated on a fixed step basis. Bruno's model generates an internally consistent salary structure after considering imposed hierarchial and budgetary constraints, other factors that should be considered in evaluating the monetary worth of an employee in addition to the two factors considered in the fixed step schedule, and the priorities and ob~ jectives of the organiza0on. The resulting salary evalua. tion model generated overcomes the drawbacks of the f'Lxedstep schedule. However, a limitation of the linear programming model is that it provides for only one priority in the objective function. To overcome this deficiency in
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the model, goal programming wlfich permits multiple objectives (or priorities) can be employed. 2. Phases of development for the mathematical programming model The early phase of development for either the linear or goal programming models is the def'mition of the components of the system by the concerned interest groups. The establishment and defmit:on of these components are directly associated with the functions of the New York State Depar~,-nent of Transportation (N.Y.S.D.O.T.) as well as to the salary schedule. The interest groups should include StaLe Administration, Civil Service Commission, N.Y.S.D. O.T. top management, the engineer's bargaining unit, and the New York State Transportation Engineers Association. This should not be co,isidered as a complete list, all other groups or individuals with valid input should be included. Job functions and individual factors of each function must be identified and properly defined in order to attain the objectives of D.O.T. The individual factors should include all those qualifications necessary for each function in the department. Each factor used ~, the job classification should contain a set of characteristics. This is done to create a salary hierarchy. 2.1. Specification o f the/ob functions
The job functions or classifications which correspond to the salary hierarchy must re;,:esent the means for full'rUing ti~e objectives of D.O.T. The department presently divides its hierarchy into the func. tions listed below (in descending order): 2 (1) Principal C;vil Engineer (Prin.) (2) Associate Civil Engineer (Asso.) (3) Senior Civil Engineer (S.E.) (4) Assi" .nt Civil Engineer (A.C.E.) (5) Junior Civil Engineer (J.E.) (6) Senior Civil Engineering Technician (S.E.T.) 2. 2. Definition o f relevant factors within functions
Seven factors were chosen as relevant by us based on our experience with the job functions and the jc~b 2 It is necessary, under New York State evil Service Law, to pass a civil service test in order to be appointed to a title. The job descriptions are stated m "Specificationsfor Positions in the New York State CivilSe~ice", publishedby the New York State Department of Civil Servicein i 976.
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F.J. Fabozzi, A.I¢. Bachner / hiP models to determine civil servicemlar~
description. In n~ ~ ay is it intended that these seven factors are all inclusive; a meeting of interest groups might well arrive at a set of different factors. The seven factors, however, are plausible. They are an extension of the two used in the fixed step salary schedule, title and years in title. The seven factors are:
Type of work area. This factor represents the degree of difficulty D.O.T. would have staffing the particular job. X2 Supervisory responsibilities of the personnel, in terms of the area or scope of responsibility. X3 The highest academic degree attained (job related) by the individual at the time of this evaluation. X4 The total number of years of work experience of the individual. This includes related experience in and out of the New York State De. partment of Transportation. Xs The total number of years of experience the individual possesses in his present title. X6 The possession of a valid New York State Professional Engineers License or Intern Engineers cert.ificate. X7 The individuals civil service rank (title, grade). Merit payments can be incorporated into the model as another factor if a measure of effectiveness can be developed and then consistently applied. However, these two requirements are generally absent and hence, this is wiry such an approach is rejected by employees and their bargaining unit.
Xt
2.3. Derivation o f relati~'e f a c w r ratings and characteristics
A relative rating scheme must be established for each factor. The ratings represent the importance of each characteristic within the job function. While a great deal of discussion might center about the specification of relative weights, there is great flexibility in the linear programming approach. As Bruno [ 1, pp. 572-573] points out. the arbitrary assignment of relative weights is not a serious limitation since the real issue, from a s~ary standpoint, is the product of the factor weight as determined by the model and the relative weight. The model considers these relative weights as well as other constraints L'~ determining the actual weights for each factor. An important concept is that a factor with too many characteristics might result .~nambiguity and a factor with too few characteristics would net sufficiently distinguish or
discriminate, from a salary standpoint, individual differences in ability or characteristics. The following relative weights are used: Factor X1 contains three classifications: easy, medium and difficult with a rating of 1, 2 and 3 respectively. Detenr~ination of these values was made by rating the Construction Section as easy (1), Planning, Traffic, etc. as wedium (2), and Design as difficult (3). Factor 3"2 contains seven classifications. The classifications are: department-wide supervisory responsibilit!es (7), ~ection-wide (6), engineer in charge (E.I.C.) of a large construction project (5), leader of a large squ~d or (E.I.C.) of a small construction project (4), leader of a sraall squad (3), responsible for a mini project (2) and no supervisory responsibilities O)Fact.or X3 contains five classifications: Ph.D. (5), Masters De~ee (4), Bavhelors Degree (3), Associates Degree (2) and High School Graduate (minimum qualificatiGn for state employment)(1). Factor X4 contains five classifications: the ratings are 12 years and up (5), eight to clever, years (4), four to ~ven years (3), two to three years (2), and zero to one year (1). Factor Xs contains five classifications: eight years and up (5), six to seven years (4), four to five years (3), two to three years (2), and zero to one year (1). Factor X6 contains three classifications: possession of a New York State Professional Engineer's License (3), Intern Engineer's Certificate (2), and no special license (1). Factor X7 contains six classifications: Principal Civil Engineer (26), Associate Civil Engineer (21), Senior Civil Engineer (17), Assistant Civil Engineer ~.14), Junior Civil Engineer (11), Senior Civil Engiaeering Technician (9). These ratings were determined by dividing the starting fixed step salary schedule by 1000 and truncating everything to the right of the quotient's decimal point 3.
3. Mathematical specification of the salary schedule The mathematical salary model utilized in this pape- was developed by Bruno [1 ]. The model is discussed below and presented in Table 1. 3 This method of determining the factor's relative weighfs is used because it is felt that the starting fixed step salary is a good indication nf *.herelative worth of each position.
F.Y. Fabozzi, A. I¢. Bachner / M P models to determine civil service salaries
Table 1 Mathematical specification of the ~ l a ~ structure 1. Theoretically highest and lowest salary for each job function:
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(26) X 3 - ~4 ~ 3700 (27) h4 - hS ~- 2700 (28) 7`S - k6 ~' 2700
Principal Civil Engineer (1) 3X 1 + 7X 2 + 5X 3 + 5X 4 + 4 X s + 3X 6 + 26X 7 = 7`1 (2) IX 1 + 6X 2 + 3X3 + 2X4 + I X s + 3,1"6 + 26,1"7 = ol
5. Factor boundaries: (29) 100 < X 1 < 1000 (30) 100 < X 2 < 1000
7,1 was restricted to be less than or equal to $32,000 since this is approximately the maximum salary in the present schedule.
(32) 100 < X 4 < 500
Associate Civil Engineer
(33) 200 < X s < 500
(3) 3X 1 + 5X 2 + 4X 3 + 5X4 + 3Xs + 3X6 + 21X7 = h2
(34) 300 < X 6 < 500
(4) 1X 1 + 3,1"2 + 3X3 + 1,1"4 + IXs + 3X6 + 21X7 = (;2
(35) 800 < X 7 < 1090
(31) 100 < X3 < 1000
Senior Civil Engineer
(5) 3X I + 3X 2 +4X 3 + 3X4 + 2X s + 3X6 + 17X 7 = k 3 (6) 1X 1 + 2X 2 + 1X 3 + 1X4 + 1X 5 + 3X6 + 17X 7 = 03
6. Budgetary requirements: (36) Bud = 538X 1 + 945X 2 + 757X 3 + 1629X 4 + 1576X s + 633X 6 + 5173X7 = $6,256,058.
Assistant Civil Engineer (7) 3X 1 + 2X 2 + 3X 3 + 3X4 + 2X s + 2X 6 + 14X7 = 7`4
(10) IX 1 + IX 2 + 2X 3 + 1X4 + IXs + IX 6 ÷ 11,1"7 = o 5
A constraint set representing the salary o f the theoretically m o s t highly qualified person (highest salary) a n d least qualified person (lowest salary) for each j o b f u n c t i o n must be developed. All salaries in this constraint set are represented 19y a linear combi-
3enior Civil Engineering Technician
nation o f the seven factors used for each j o b function.
(11) 3X 1 + IX 2 + 2X 3 + 2X4 4. 3X 5 + IX 6 + 9X 7 = h6
The coefficients for each j o b f u n c t i o n represent the highest (lowest) rated characteristic associated with the correspondin~ factor for the theoretically most highly (least) qualified person. The theoretically highest and lowest salary for j o b f u n c t i o n / w i l l be denoted b y ~,/and oi respectively. There are six j o b functions in the m o d e l ; therefore, there will be twelve constraints. The d e t e r m i n , . tion o f relevant factors and characteristics was decided u p o n in accordance with current D e p a r t m e n t o f T r a n s p o r t a t i o n policies and qualification criteria. The theoretically highest and lowest salary for each j o b f u n c t i o n are given b y ( 1 ) - ( 1 2 ) in Table 1. The entire set o f job f u n c t i o n constraints are equalities. The solution to the ~,/s and o / s will be produced through the model. The optim~d values generated f r o m the m o d e l can t h e n be c o m p a r e d with the m a x i m u m and m i n i m u m salary for each j o b f u n c t i o n u n d e r the present salary "~chedule. A condition i m p o s e d b y the e m p l o y e e bargaining unit for i m p l e m e n t i n g the optimal salary schedule generated b y the m a t h e m a t i c a l p r o g r a m m i n g m o d e l would p r o b a b l y be that the o p t i m a l salaries for each j o b f u n c t i o n m a y not be less t h a n the present range for the j o b f u n c t i o n . If this c o n d i t i o n fails to be met, a
(8) IX 1 + 2X 2 + 1X 3 + 2X4 + IX s + 1X6 + 14X7 = o 4 Junior Civil Engineer (9) 3X 1 + 2,1"2 + 3X 3 + 3X4 + 2X s + IX 6 + 1 IX 7 = 7,S
(12) I , " I ~ . I X 2 + I X 3 + l X
4
1Xs+IX6+9XT=o6
2. Minimum fraction salary spread within each job function (13) o I ~ 0.83 ~1 (14) 02 ;) 0.82 7,2 (15) 03 ;) 0.81 7`3 (16) 04 ;) 0.81 h4 (17) o 5 ~ 0.80 7`5 (18) 06 ~ 0.79 7,6
3. Minimum fraction salary overlap between job functions: (19) X2 ~ 0.93 o 1 (20) 7,3 ;) 0.96 02 (21) k4 ~ 1.00 03
(22) 7,S ;) 1.dO 0 4 (23) 7,6 ;) 1.00 o s 4. Minimum dollar spread between the highest salary for each job function: (24) 7,1 - X2 ~ 5700 (25) k2 - x3 ;~ 4800
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F.Z Fabozzi, A. W. Bachner / MP models to determine civil service salaries
lower bound can be placed on the job functions which far to meet tl~is condRion. The agency, on the other hand, might seek an upper bound on the minimum and maximum value for some job functions if they differ substantially from the present range. This demand by any particular agency would probably be requested to keep the salary schedule in line with that of similar job functions at comparable agencies. An enviroranental constraint set is formulated to reflect the rel~tionships desired within and between each job function. The relationships that are con. sidered are the salary spread within each job function, overlaps in salary hierarchy, and the salary spread between the highest salary in each job function. These constraints are represented by (13)-(28) in Table 1. In order to ensure that no single factor dominates the salary schedule, upper and lower bounds should be placed on each factor 4. In this application, however, the boundaries were set after discussing each factor's monetary value with ~-l.ployees in the Department. In applying the moael, the negotiating unit for the employees must make a more extensive survey of its members in order to a~certain the consensus of opinion with respect to the monetary worth of each factor. The same process should be used to determine ~vhat factors should be incorporated into the salary evaluation model. The bargaining team for tl:e agency under tl~e direction of the Civil Service Commission will also establish factors that it considers relevant and the range for the renumetation for each factor. The final factors included, as well as the compensation range for each factor, will then depend upon the bargaining ability of each of the bargai~fingagents. The boundaries for each of the seven factors are given by (29)-(35) in Table 1. The most important constraint (inherent in all salary schemes) is the budget. The Department's resources are fixed and cannot be exceeded; if,ere. fore, the sum of all the salaries must be equal to the Department's salary budget. The budgetary constraint is. given in Table 1 by (36). The necessary data for the coefficient for each factor in the budgetary constraint was obtained from perso,,mel records. Each coefficient repregcnts the weighted total of the number of employees possessing each characteristic for a factor using the relative rating as the weight. 4 The boundariescan be parameterized to observe the effect on the salary schedule.
Due to confidentiality of salaries, an exact budget figure could not be obtained for this model. The figure employed i.n the model is an estimate of the Department's resources. It was obtained by estimating each employee's salary using the fLxed step salary schedule and the employee's tLme in title.
4. The linear programming model In the linear programming formulation of the salary evaluation model, conditions (1)-(36) become constraints imposed on the model. The objective function then reflects the highest priority or objective of the Department. However, the employees may object to the objective set by the Department. Consequently, the employees may seek an alternative objective to be optimized. The salary evaluation model can then be solved for each objective function. Based on the differences in the optimal values obtained using each objective function, compromises cart be made. These compromises can then be incorporated into the model by imposw,g the necessary constraint(s). For illustrative purposes the objective function used in this application of the model will reflect the priority of maximizing the "staffing factor" (Xl). This is expressed mathematically as: Maximize: XI. As previously mentioned, this crherion could reflect D.O.T.'s priority or objective of enticing engineers to accept or remain in positions that are considered lackluster. Other possible objective functions v/ill be discussed below. The optimal weight for each factor in the salary scheme was determined by solving the linear programming model s that is, maximizing XI subject to the constraint set given by (1)-(36). Table 2 summarizes these optimal weights. The theoretically highest and lowest salary for each job classification were then determined by substituting these optimal weights into equations 1 through 12 of the salary model. Table 3 summarizes the salaries for each of the job classifications for the Department and compares the range with the maximum and minimum under the prevailing fixed step schedule. Notice that the maximum and minimum optimal salaries exceed those of the present schedule. Had this not occurred and the interested parties wanted : ~ ensure $ IBM LinearProgrammingSystem/360 (LPS/360) (360ACO-IEX) was utilized in obtainingthe solution to this model.
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F.Z Fabozzi, A. W. Bachner / M P models to determine civil service salaries
Table 2 Optimal factor weights for alternative objective functions using linear and goal prograr~:ming a Factor
Xl )/'2 X3 X4 XS X6 X7
Linear programming results
Staff'mg difficulty Supervisory responsibilities Academic degree Workext~erience Workexperience in title Professional engineer's lie. Civil service rank
Goal programming results
Maximize
Maximize
Maximize
Xl
X2
Xs
Maximize b X 1 and h5
Maximize c X 1 and X5
$623 542 544 207 200 300 803
$608 576 516 211 203 300 800
$600 488 600 184 200 300 814
$616 525 562 201 200 300 806
$605 497 588 190 200 300 812
a Roundea to nearest dollar.
b Equal weight given to both variables in the highest prioriW. c Greater weight given to maximization of k s relative to X 1 in the highest priority. Table 3 Optimal salary schedule for alternative objective functions using linear and goal programming a Job function
1 Principal Civil Engineer
(highest) Oowest) AssociateCivil Engineer (highest) (lowest) Senior Civil Engineer (highest) (lowest) AssistantCivil Engineer (highest) (lowest) Junior Civil Engineer (highest) (lowest) Senior Engineering Tech. (highest) (lowesG
2 3 4 5 6
Linear programming results
Goal programming results
Present d
Maximize Maximize Maximize XI X2 XS
Maximize b X 1 and 7,5
Maximizec X 1 and X5
$31,940 26,516 26,255 21,545 21,467 17,429 17,544 14,142 14,151 11,377 11,405 9,029
$32,000 27,900 26,156 22,050 21,244 17,208 17,449 14,407 14,741 11,793 12,041 9,643
$32,000 27,918 26,156 22,110 21,279 17,237 17,474 14,418 14,755 il,835 12,055 9,661
$32,000 27,944 26,157 22,202 21,334 17,281 17,513 14,435 14,797 11,900 12,077 9,688
$32,000 27,935 26,130 21,996 21,152 17,189 17,36 ! I4,401 14,661 11,729 11,961 9,613
$32,000 27,959 26,154 22,241 21,354 17,297 17,524 14,439 14,783 11,925 12,083 9,698
See footnotes a, b and c on Table 2. d Source: New York State Civil Service Law, Section 1308 131 (1976).
its occurrence, constraints (2) through (12) would have specified minimum values 6. To calculate an individual salary, the individual's relative ratings for the particular characteristics he possesses are multiplied by the factor weights. i
Yi/Xi = S
where S = the individual's salary; Yii = the relative rating of characteristic ] found in factor i which is 6 ',~2ten ?-1 was unrestricted it took on the value of $32,530.
possessed by the individual; Xi = the weight for factor i (given by the solution set). For example, the salary for an Assistant Civil Engineer (14X7), working in Planning (2X1), with supervisory responsibilities of a mini project or "job" (2X2), with an Associates Degree (2Xa), three years of work experience (2X4), one year of experience in title (iXs) and an Intern Engineer's Certificate (2X6) is $15,876. Under the fixed step salary schedule this employee would receive a salary of $14.142. Of course, not all salaries will be greater than that existing under the present schedule. Whether compensation
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is more or less depends on the relat~e rating of an individual. If additional resources became available, the linear progranuning model can incorporate the increase into the salary schedule in a logical and consistent manner. Table 4 lists the salary schedules which result from increasin£ the Department's resources of $6,256,058 by 10%, 20%, and 30%. In the generation of alternative salasy structures over time as additional reumrces are made available and the objective function change, it would be possible for an individual salary to decline. The problem is further compounded when new personnel are recruited and present personnel change the characteristics they possess. Such a system of compensation in which one individual's salary declines while a co-worker's salary increases would be totally unacceptable to most employees. However, this problem can be overcome by simply requiring that the optimal weight for each factor in the existing schedule be set as a lower limit when revising the model. This will ensure that no salary will decline. The model optimized the "staff'mg difficulties" factor (Xt) but another factor or salary could also serve as the objective function. For example, the Department might desire to optimize the "supervisory responsibility" factor (,I"2). Optimizing (,!"2) would give employees additional incentive to seek out and accept responsibilities. The optimal weights for each factor in the salary scheme if .I"2 is maximized subject to (1)-(36) is given in Table 2. The optimal salary for each job function is given in Table 3. An example of a salary used as an objective rune-
adarla
tion would be the maximization of the Junior En. gineer's salary. If the Department demed to retain these highly skilled technicians Xs, highest Junior Civil Engineer, should be maximized. Table 2 summarizes these optimal weights and Table 3 provides the salaries for the theoretically high and low for each job function when Xs is maximized. These salaw schedules demonstrate the versatifity of the linear programming model.
S. The goal programming model The deficiency of the linear programming model lies in its inability to assimilate multiple conflicting objectives or priorities. Goal programming, a modification and extension of linear programming, permits such flexibility. For example, if the Department seeks to maximize both the value of the "staff'mg factor" and the salary of the theoretically highest qualified Junior Civil Engineer, the linear program. ming model cannot handle tiffs problem whereas goal programming can. Moreover, if the employees disagree with the objective function set by the Department and seek an alternative objective function, the goal programming model can be used since it can con. sider both objective functions simultaneously. In goal programming each of the conditions (1)-(36) in Table 1 are considered goals and referred to as goal constraints. The objective function of a goal programming model differs from that of linear programming in that it does not contain the decision variables (i.e. factors) in the objective function.
Table 4 Optimal salary schedule with additional resources: XI maximized Prin.
(highest) 0owest) Asso. (highest) (lowest) S.E. (highest) 0owest) A.C.E. (highest) OowesO J.E. (highest) (lowest) S.E.T. (highest) (lowest) Depar~'nent resources
$32,000 27,903 26,153 22,049 21,241 17,205 17,444 14,404 14,737 11,790 12,037 9,640
$35,454 31,136 29,063 24,813 23,890 19,590 19,786 16,283 16,679 13,343 13,910 10,989
$6,256,058
$6,~82,000
a The con~tr~nt that 7,1 be equal :o $32,000 was eli~hlated.
$38,996 a 34,242 31,926 27,256 26,269 21,403 21,617 17,681 18,206 14,565 15,065 11,952 $7,507,000
$41,221 35,679 33,566 28,251 27,262 22,082 22,303 18,262 18,803 15,043 15,611 12,333 $8,133,000
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F.J. Fabozzi, A.I¢. Bachner / M P models to determine civil service salaries
Instead, negative and/or positive deviational variables are assigned to each goal constraint depending upon whether underachievement or overachievement or both are allowed. Each deviational variable then appears in the objective function according to the priority level assigned to R 7. Goal programming then seeks to minimize the value of the deviational variables. In this application, maximization of the "staff'mg factor" (XI) and the salary of the theoretically highest qualified Junior Civil Engineer Q,s) are assumed. Within a given priority level, different weights can be assigned to a goal s. Tile goal programming model is discussed below. The constraint goals for the theoretically maximum and minimum salary for each job function ( 2 ) - ( 1 2 ) remain as equations in the goal programming model 9. Neither pesitive nor negative deviational variables are introcuced; that is, underachievement or overachievement is not permitted. Similarly, the budgetary goal constraint (36) must be met exactly. The goal constraints ( 1 3 ) - ( 2 8 ) can be rewritten as "greater than cr equal to" goal constraints by simply subtracting the value on the right hand side from both sides of the inequality. Since each goal constraint of the "greater than or equal to" type permits overachievement of a goal, a positive dev. ational variable (d~ for i = 13, 14, ..., 28) is included in each of the respective goal constraints. The goal constraints (29)-(35) which provide monetary bounds for the 7 According to Lee 14, p.27]: "in order to achieve the ordinal solution - that is, to achieve the goals according to their importance - negative and/or positive deviations about the goal must be ranked according to the 'preemptive' priority factors. In this way the low-order goals are considered only after higher-order goals are achieved as desired. If there are goals in k ranks, the 'preemptive' priority factor Pi (] -" 1, 2..... k) should be assigned to the negative and/or positive devlational variables. The 'preemptive' priority factors have the relationship ofP/>>> P]+ 1, which implies that the multip"--_lion of n, however large it may be, cannot make P]+ 1 greater than or equal to Pi"" 8 Within a given priority level, if two or more goals exist, weights must be assigned. Conjoint measurement, a technique developed by mathematical psychologists to quantify judgmental data, could be employed. Using conjoint measures, ranked ordered goals within a preemptive priority level would yield interval-sealed output that can be used as weights. For a primarily expository article on this subject see Green and Rao [ 3 l. 9 The first constraint limited h I tO be less than or equal to $32,000. Hence, this type of constraint will have a negative deviational goal (arT).
factors can be divided Lnto "less than or equal to" constraints for the upper and lower bounds respectively. Negative deviational variables (d/-tj for iU = 29, 30, ..., 35) are assigned to the goal constraints for the upper bounds while positive deviational variables (d~/g for iL = 29, 30, ..., 35) are included in the lower bound goal constraints. Deviational variables which permit the maximization of X1 and Xs must be incorporated into the goal programming model. This is accomplished by the following two goal constraints: (37) XI + d37 = c~ and (38) Xs + d~8 = ~3 where d~7 and d~8 are negative deviational variables and a and/3 are arbitrarily assigned values which are high enough so that they cannot be achieved. The objective function which requires the minlmization of the deviational variables will ensure that X1 and Xs are maximized. The objective function for the goal programming model is then, minimize: ....
SL, d go, d
d3oL, ...,
ou, ..., d 5o, d D
where Pl is tile first (highest) pr~c,rity and 1"2 is tile second (and in tiffs case lowest) priority 10 Two goal programming results are presented in Tables 2 and 3. The first set of optimal val,~es reported is based on the assumption that both goals within the highest priority (i.e. maximization of Xl and X~) are given equal weight. However, as indicated above, different weights can be assigned within a priority level. The second goal programming result is still based on maximizing both XI and Xs as the highest priority, yet in this formulation maximizing Xs is considered to be more important "". The weights assigned to the importance of a variable within a priority level would be set by the Department based on its own objectives assuming the employees do not contest the objectives. In a case where the objectives of the interested parties conflict
10 The computer algorithm to solve this problem is ~ven by Lee [4]. 11 Specifically, the maximization of h s was considered three times more important than the maximization of X 1 for illustration purposes in this formulation.
198
F~I. Fabozzi, A.I¢. Bachner / MP models to determine cidl zervic¢
and both seek that their objective be in the highest priority, then a true compromise position can be attained by using equal weights for both objectives in the highest priority. Depmmres from this weighting scheme depend on the negotiating ability of the interested parties.
6. Conclusion The mathematical approach to salary evaluation permits a consideration of a virtu~y unlimited number of pertinent factors for a job function, the f'mancial resources available to the organization, and the establis~ed objectives of the organization in constructing a salary schedule. The linear programming model of Bruno permits one objective to be considered while the goal programming model allows multiple conflicting objectives. Both models generate salary schedules which overcome the criticisms of the rigid fLxed step salary schedule traditionally used. By parameterizing the bounds for the factor values, the organization can assess alternative salary schedules and determine the necessary budget. Information of this type would be valuable in analyzing the cost of demands made by the bar~at'ning unit representing the civil servants. Moreo';er, the dual solution to the optimal value provide the organization with flexibility :n negotiation since the dual solu. tion indicates the unit costs associated with forcing an incremental doV ,r into the salary weight for each factor. Parameterizing the f'mancial resources avail. able indicates how any increase in funds would be distributed to each factor. This eliminates "across the board" increases that generally accompany the decision to provide additior~al funds. This would not result in a decline in salary for any individual even if
the objective function is changed as long as the optimal monetary value for each factor in the prior period are established as minimum values in subsequent negotiations. There are numerous difficulties that must be surmounted before such models can obtain greater acc¢ptance in helping to determine salary schedules. First, the political power of the bargaining units ou',weigh systems of rationality in issues of selfinterest. Second, many unions seldom encourage or permit significant differentials between members lest their plfilosophy of"all-for-one.and-one-for.alr' be shattered. Finally, Ore programming approach assumes that there are individt~als capable of using such models, patient enough to work out the factors and agree on their relative weights a~,d sincere enough to worry about changing a traditionally accepted form of establishing remuneration for civil servants.
References [ 1] I.E. Bruno, Compensationof school district personnel, Management Sci. 17 (1971) 569-587. [2] A. Chames, W.W.Cooper and R.O. Ferguson,Optimal estimation of executivecompensahon by linear programruing, ManagementSci. 1 (1955) 138-151. [3] P.A. Green and V.R. Rao, Conjoint measurementfor quantifyingjudgmental data, J. MarketingRes. 8 (1971) 355-360. [41 S.M. Lee, Goal Programmingfor Decision Analysis (Auerbach Publishers,Philadelphia,PA, 1972). [5] J. Lennon, Lennor,rebuts radio editorial on merit, Civil Service Leader (February 11, 1977). [6] F.P. Rehmusand H.M. Wagner, Applyinglinear programruing to your pay structure, BusinessHorizons4 (1963) 89-98. [7] J.V. Smith, Merit compensation: The ideal and reality, Personnel J. 51 (May 1972). [8] A. Zack, Meeting the risingcost of public sector settlements, Monthly Labor Rev. 96 (May 1973).
F r a n k J. F A B O Z Z I , School of Business, Hofs~-a Un~ersity, Hempstead, N Y 11550, U.S.A. A l f r e d W. B A C H N ~ ! R New York State Departm~.nt of Transportation, New York, NY, U.S.A. Received 19 October 1977 Revised 3 April 1978 Increased criticism has been directed in recent years at the present fixed .qtep salary method used for compensating civil servants. Under this method the relative monetary value of an individual t~ he department or agency is bated only on two factors: civil service grade and the number of years of experience in the particular grade. This paper describes how two mathematical programming methods, linear pro~amming and goal programming, can be applied to determine a civil service salary structure for the New York State Department of Transportation (Region 10). The mathematical salary model employed was developed by J.E. Bruno. I, Introduction In the civil service the salary compensation of each employee is open, and it is therefore of primary importance that each employee feels that he or she is being not only adequately compensated, but also fairly compensated when compared to fellow employees, their duties, responsibilities, and expertise. We wish to thank the two referees of this journal, Clive Purkiss and Kees Verhoeven, and Lesley Browder for several helpful criticisms and suggestions. John Pizzariella provided valuab~_eprogramming assistance. Of course, the ~xaditional caveat concerning .~ny remaining errors applies. The views expressed in this paper represent those of the author. Salaries are a delicate matter, even among mature adults. They satisfy ego and gwe the worker prestige besides satisfying economic needs. A poorly designed (open or secret) salary eomper, sation system can only create misunderstanding, frustration and low morale. There is a great deal of discontent among civil
1 Historically, the fixed step salary schedule was implemented to guarantee civil servants, who were prohibited from joining collective bargaining units (unions), future salary raises [8, p.40]. When legislation mandated the civil servant to be represented by a bargaining unit, the fixed step salary schedule was a retained vestige; perhaps this was due to employee lack of confidence in the early bargaining units.
© North-Holland Publishing Company European Journai of Operational Research 3 (!979) i 90-198. 190
F.Z Fabozzi, A. W. Bachner / MP models to determine civil service ~alaries
The "merit system" employed in the private sector has been proposed as an alternative compensation system for the public sector. This system ~rovides for salary movements based on a set of objectives established and reviewed by management. The "merit system" is strongly supported by Administration and opposed by the Civil Service Employees Association (CSEA) [51. The "merit system" requires the development of objectives and the necessary criteria for evaluation. Herein lies the difficulty. There are presently no acceptable methods for measuring the effectiveness of many civil service employees. The evaluation then becomes qualitative and highly subjective leaving no recourse for debate. It is this qualitative and subjective evaluation that repulses the CSEA. In addition, the inability of the employee to duplicate the administration's evaluation compounds the employee's sense of frustration and defeat [7], and this detriment disqualifies the "merit system" as a plausible salary scheme by itself. However, it can be incorporated as one factor in the salary evaluation model described in this paper. The purpose of this paper is to demonstrate how mathematical programming can be used to develop a canpensation system which takes into consideration several characteristics of a job function and also reflects the objectives of the department or agency. The model will be applied to the New York State Department of Transportation, Region 10, where the traditional fixed step salary schedule is tl:e present method of salary evaluation for engineering personnel. Clmrnes et al. [2] and Relunus and Wagner [6] have suggested that linear programming could be used to determine executive compensation. Recently, Bruno [ 1] ires developed a linear programming salary evaluation model to compensate school district personnel who are presently compensated on a fixed step basis. Bruno's model generates an internally consistent salary structure after considering imposed hierarchial and budgetary constraints, other factors that should be considered in evaluating the monetary worth of an employee in addition to the two factors considered in the fixed step schedule, and the priorities and ob~ jectives of the organiza0on. The resulting salary evalua. tion model generated overcomes the drawbacks of the f'Lxedstep schedule. However, a limitation of the linear programming model is that it provides for only one priority in the objective function. To overcome this deficiency in
191
the model, goal programming wlfich permits multiple objectives (or priorities) can be employed. 2. Phases of development for the mathematical programming model The early phase of development for either the linear or goal programming models is the def'mition of the components of the system by the concerned interest groups. The establishment and defmit:on of these components are directly associated with the functions of the New York State Depar~,-nent of Transportation (N.Y.S.D.O.T.) as well as to the salary schedule. The interest groups should include StaLe Administration, Civil Service Commission, N.Y.S.D. O.T. top management, the engineer's bargaining unit, and the New York State Transportation Engineers Association. This should not be co,isidered as a complete list, all other groups or individuals with valid input should be included. Job functions and individual factors of each function must be identified and properly defined in order to attain the objectives of D.O.T. The individual factors should include all those qualifications necessary for each function in the department. Each factor used ~, the job classification should contain a set of characteristics. This is done to create a salary hierarchy. 2.1. Specification o f the/ob functions
The job functions or classifications which correspond to the salary hierarchy must re;,:esent the means for full'rUing ti~e objectives of D.O.T. The department presently divides its hierarchy into the func. tions listed below (in descending order): 2 (1) Principal C;vil Engineer (Prin.) (2) Associate Civil Engineer (Asso.) (3) Senior Civil Engineer (S.E.) (4) Assi" .nt Civil Engineer (A.C.E.) (5) Junior Civil Engineer (J.E.) (6) Senior Civil Engineering Technician (S.E.T.) 2. 2. Definition o f relevant factors within functions
Seven factors were chosen as relevant by us based on our experience with the job functions and the jc~b 2 It is necessary, under New York State evil Service Law, to pass a civil service test in order to be appointed to a title. The job descriptions are stated m "Specificationsfor Positions in the New York State CivilSe~ice", publishedby the New York State Department of Civil Servicein i 976.
192
F.J. Fabozzi, A.I¢. Bachner / hiP models to determine civil servicemlar~
description. In n~ ~ ay is it intended that these seven factors are all inclusive; a meeting of interest groups might well arrive at a set of different factors. The seven factors, however, are plausible. They are an extension of the two used in the fixed step salary schedule, title and years in title. The seven factors are:
Type of work area. This factor represents the degree of difficulty D.O.T. would have staffing the particular job. X2 Supervisory responsibilities of the personnel, in terms of the area or scope of responsibility. X3 The highest academic degree attained (job related) by the individual at the time of this evaluation. X4 The total number of years of work experience of the individual. This includes related experience in and out of the New York State De. partment of Transportation. Xs The total number of years of experience the individual possesses in his present title. X6 The possession of a valid New York State Professional Engineers License or Intern Engineers cert.ificate. X7 The individuals civil service rank (title, grade). Merit payments can be incorporated into the model as another factor if a measure of effectiveness can be developed and then consistently applied. However, these two requirements are generally absent and hence, this is wiry such an approach is rejected by employees and their bargaining unit.
Xt
2.3. Derivation o f relati~'e f a c w r ratings and characteristics
A relative rating scheme must be established for each factor. The ratings represent the importance of each characteristic within the job function. While a great deal of discussion might center about the specification of relative weights, there is great flexibility in the linear programming approach. As Bruno [ 1, pp. 572-573] points out. the arbitrary assignment of relative weights is not a serious limitation since the real issue, from a s~ary standpoint, is the product of the factor weight as determined by the model and the relative weight. The model considers these relative weights as well as other constraints L'~ determining the actual weights for each factor. An important concept is that a factor with too many characteristics might result .~nambiguity and a factor with too few characteristics would net sufficiently distinguish or
discriminate, from a salary standpoint, individual differences in ability or characteristics. The following relative weights are used: Factor X1 contains three classifications: easy, medium and difficult with a rating of 1, 2 and 3 respectively. Detenr~ination of these values was made by rating the Construction Section as easy (1), Planning, Traffic, etc. as wedium (2), and Design as difficult (3). Factor 3"2 contains seven classifications. The classifications are: department-wide supervisory responsibilit!es (7), ~ection-wide (6), engineer in charge (E.I.C.) of a large construction project (5), leader of a large squ~d or (E.I.C.) of a small construction project (4), leader of a sraall squad (3), responsible for a mini project (2) and no supervisory responsibilities O)Fact.or X3 contains five classifications: Ph.D. (5), Masters De~ee (4), Bavhelors Degree (3), Associates Degree (2) and High School Graduate (minimum qualificatiGn for state employment)(1). Factor X4 contains five classifications: the ratings are 12 years and up (5), eight to clever, years (4), four to ~ven years (3), two to three years (2), and zero to one year (1). Factor Xs contains five classifications: eight years and up (5), six to seven years (4), four to five years (3), two to three years (2), and zero to one year (1). Factor X6 contains three classifications: possession of a New York State Professional Engineer's License (3), Intern Engineer's Certificate (2), and no special license (1). Factor X7 contains six classifications: Principal Civil Engineer (26), Associate Civil Engineer (21), Senior Civil Engineer (17), Assistant Civil Engineer ~.14), Junior Civil Engineer (11), Senior Civil Engiaeering Technician (9). These ratings were determined by dividing the starting fixed step salary schedule by 1000 and truncating everything to the right of the quotient's decimal point 3.
3. Mathematical specification of the salary schedule The mathematical salary model utilized in this pape- was developed by Bruno [1 ]. The model is discussed below and presented in Table 1. 3 This method of determining the factor's relative weighfs is used because it is felt that the starting fixed step salary is a good indication nf *.herelative worth of each position.
F.Y. Fabozzi, A. I¢. Bachner / M P models to determine civil service salaries
Table 1 Mathematical specification of the ~ l a ~ structure 1. Theoretically highest and lowest salary for each job function:
193
(26) X 3 - ~4 ~ 3700 (27) h4 - hS ~- 2700 (28) 7`S - k6 ~' 2700
Principal Civil Engineer (1) 3X 1 + 7X 2 + 5X 3 + 5X 4 + 4 X s + 3X 6 + 26X 7 = 7`1 (2) IX 1 + 6X 2 + 3X3 + 2X4 + I X s + 3,1"6 + 26,1"7 = ol
5. Factor boundaries: (29) 100 < X 1 < 1000 (30) 100 < X 2 < 1000
7,1 was restricted to be less than or equal to $32,000 since this is approximately the maximum salary in the present schedule.
(32) 100 < X 4 < 500
Associate Civil Engineer
(33) 200 < X s < 500
(3) 3X 1 + 5X 2 + 4X 3 + 5X4 + 3Xs + 3X6 + 21X7 = h2
(34) 300 < X 6 < 500
(4) 1X 1 + 3,1"2 + 3X3 + 1,1"4 + IXs + 3X6 + 21X7 = (;2
(35) 800 < X 7 < 1090
(31) 100 < X3 < 1000
Senior Civil Engineer
(5) 3X I + 3X 2 +4X 3 + 3X4 + 2X s + 3X6 + 17X 7 = k 3 (6) 1X 1 + 2X 2 + 1X 3 + 1X4 + 1X 5 + 3X6 + 17X 7 = 03
6. Budgetary requirements: (36) Bud = 538X 1 + 945X 2 + 757X 3 + 1629X 4 + 1576X s + 633X 6 + 5173X7 = $6,256,058.
Assistant Civil Engineer (7) 3X 1 + 2X 2 + 3X 3 + 3X4 + 2X s + 2X 6 + 14X7 = 7`4
(10) IX 1 + IX 2 + 2X 3 + 1X4 + IXs + IX 6 ÷ 11,1"7 = o 5
A constraint set representing the salary o f the theoretically m o s t highly qualified person (highest salary) a n d least qualified person (lowest salary) for each j o b f u n c t i o n must be developed. All salaries in this constraint set are represented 19y a linear combi-
3enior Civil Engineering Technician
nation o f the seven factors used for each j o b function.
(11) 3X 1 + IX 2 + 2X 3 + 2X4 4. 3X 5 + IX 6 + 9X 7 = h6
The coefficients for each j o b f u n c t i o n represent the highest (lowest) rated characteristic associated with the correspondin~ factor for the theoretically most highly (least) qualified person. The theoretically highest and lowest salary for j o b f u n c t i o n / w i l l be denoted b y ~,/and oi respectively. There are six j o b functions in the m o d e l ; therefore, there will be twelve constraints. The d e t e r m i n , . tion o f relevant factors and characteristics was decided u p o n in accordance with current D e p a r t m e n t o f T r a n s p o r t a t i o n policies and qualification criteria. The theoretically highest and lowest salary for each j o b f u n c t i o n are given b y ( 1 ) - ( 1 2 ) in Table 1. The entire set o f job f u n c t i o n constraints are equalities. The solution to the ~,/s and o / s will be produced through the model. The optim~d values generated f r o m the m o d e l can t h e n be c o m p a r e d with the m a x i m u m and m i n i m u m salary for each j o b f u n c t i o n u n d e r the present salary "~chedule. A condition i m p o s e d b y the e m p l o y e e bargaining unit for i m p l e m e n t i n g the optimal salary schedule generated b y the m a t h e m a t i c a l p r o g r a m m i n g m o d e l would p r o b a b l y be that the o p t i m a l salaries for each j o b f u n c t i o n m a y not be less t h a n the present range for the j o b f u n c t i o n . If this c o n d i t i o n fails to be met, a
(8) IX 1 + 2X 2 + 1X 3 + 2X4 + IX s + 1X6 + 14X7 = o 4 Junior Civil Engineer (9) 3X 1 + 2,1"2 + 3X 3 + 3X4 + 2X s + IX 6 + 1 IX 7 = 7,S
(12) I , " I ~ . I X 2 + I X 3 + l X
4
1Xs+IX6+9XT=o6
2. Minimum fraction salary spread within each job function (13) o I ~ 0.83 ~1 (14) 02 ;) 0.82 7,2 (15) 03 ;) 0.81 7`3 (16) 04 ;) 0.81 h4 (17) o 5 ~ 0.80 7`5 (18) 06 ~ 0.79 7,6
3. Minimum fraction salary overlap between job functions: (19) X2 ~ 0.93 o 1 (20) 7,3 ;) 0.96 02 (21) k4 ~ 1.00 03
(22) 7,S ;) 1.dO 0 4 (23) 7,6 ;) 1.00 o s 4. Minimum dollar spread between the highest salary for each job function: (24) 7,1 - X2 ~ 5700 (25) k2 - x3 ;~ 4800
194
F.Z Fabozzi, A. W. Bachner / MP models to determine civil service salaries
lower bound can be placed on the job functions which far to meet tl~is condRion. The agency, on the other hand, might seek an upper bound on the minimum and maximum value for some job functions if they differ substantially from the present range. This demand by any particular agency would probably be requested to keep the salary schedule in line with that of similar job functions at comparable agencies. An enviroranental constraint set is formulated to reflect the rel~tionships desired within and between each job function. The relationships that are con. sidered are the salary spread within each job function, overlaps in salary hierarchy, and the salary spread between the highest salary in each job function. These constraints are represented by (13)-(28) in Table 1. In order to ensure that no single factor dominates the salary schedule, upper and lower bounds should be placed on each factor 4. In this application, however, the boundaries were set after discussing each factor's monetary value with ~-l.ployees in the Department. In applying the moael, the negotiating unit for the employees must make a more extensive survey of its members in order to a~certain the consensus of opinion with respect to the monetary worth of each factor. The same process should be used to determine ~vhat factors should be incorporated into the salary evaluation model. The bargaining team for tl:e agency under tl~e direction of the Civil Service Commission will also establish factors that it considers relevant and the range for the renumetation for each factor. The final factors included, as well as the compensation range for each factor, will then depend upon the bargaining ability of each of the bargai~fingagents. The boundaries for each of the seven factors are given by (29)-(35) in Table 1. The most important constraint (inherent in all salary schemes) is the budget. The Department's resources are fixed and cannot be exceeded; if,ere. fore, the sum of all the salaries must be equal to the Department's salary budget. The budgetary constraint is. given in Table 1 by (36). The necessary data for the coefficient for each factor in the budgetary constraint was obtained from perso,,mel records. Each coefficient repregcnts the weighted total of the number of employees possessing each characteristic for a factor using the relative rating as the weight. 4 The boundariescan be parameterized to observe the effect on the salary schedule.
Due to confidentiality of salaries, an exact budget figure could not be obtained for this model. The figure employed i.n the model is an estimate of the Department's resources. It was obtained by estimating each employee's salary using the fLxed step salary schedule and the employee's tLme in title.
4. The linear programming model In the linear programming formulation of the salary evaluation model, conditions (1)-(36) become constraints imposed on the model. The objective function then reflects the highest priority or objective of the Department. However, the employees may object to the objective set by the Department. Consequently, the employees may seek an alternative objective to be optimized. The salary evaluation model can then be solved for each objective function. Based on the differences in the optimal values obtained using each objective function, compromises cart be made. These compromises can then be incorporated into the model by imposw,g the necessary constraint(s). For illustrative purposes the objective function used in this application of the model will reflect the priority of maximizing the "staffing factor" (Xl). This is expressed mathematically as: Maximize: XI. As previously mentioned, this crherion could reflect D.O.T.'s priority or objective of enticing engineers to accept or remain in positions that are considered lackluster. Other possible objective functions v/ill be discussed below. The optimal weight for each factor in the salary scheme was determined by solving the linear programming model s that is, maximizing XI subject to the constraint set given by (1)-(36). Table 2 summarizes these optimal weights. The theoretically highest and lowest salary for each job classification were then determined by substituting these optimal weights into equations 1 through 12 of the salary model. Table 3 summarizes the salaries for each of the job classifications for the Department and compares the range with the maximum and minimum under the prevailing fixed step schedule. Notice that the maximum and minimum optimal salaries exceed those of the present schedule. Had this not occurred and the interested parties wanted : ~ ensure $ IBM LinearProgrammingSystem/360 (LPS/360) (360ACO-IEX) was utilized in obtainingthe solution to this model.
195
F.Z Fabozzi, A. W. Bachner / M P models to determine civil service salaries
Table 2 Optimal factor weights for alternative objective functions using linear and goal prograr~:ming a Factor
Xl )/'2 X3 X4 XS X6 X7
Linear programming results
Staff'mg difficulty Supervisory responsibilities Academic degree Workext~erience Workexperience in title Professional engineer's lie. Civil service rank
Goal programming results
Maximize
Maximize
Maximize
Xl
X2
Xs
Maximize b X 1 and h5
Maximize c X 1 and X5
$623 542 544 207 200 300 803
$608 576 516 211 203 300 800
$600 488 600 184 200 300 814
$616 525 562 201 200 300 806
$605 497 588 190 200 300 812
a Roundea to nearest dollar.
b Equal weight given to both variables in the highest prioriW. c Greater weight given to maximization of k s relative to X 1 in the highest priority. Table 3 Optimal salary schedule for alternative objective functions using linear and goal programming a Job function
1 Principal Civil Engineer
(highest) Oowest) AssociateCivil Engineer (highest) (lowest) Senior Civil Engineer (highest) (lowest) AssistantCivil Engineer (highest) (lowest) Junior Civil Engineer (highest) (lowest) Senior Engineering Tech. (highest) (lowesG
2 3 4 5 6
Linear programming results
Goal programming results
Present d
Maximize Maximize Maximize XI X2 XS
Maximize b X 1 and 7,5
Maximizec X 1 and X5
$31,940 26,516 26,255 21,545 21,467 17,429 17,544 14,142 14,151 11,377 11,405 9,029
$32,000 27,900 26,156 22,050 21,244 17,208 17,449 14,407 14,741 11,793 12,041 9,643
$32,000 27,918 26,156 22,110 21,279 17,237 17,474 14,418 14,755 il,835 12,055 9,661
$32,000 27,944 26,157 22,202 21,334 17,281 17,513 14,435 14,797 11,900 12,077 9,688
$32,000 27,935 26,130 21,996 21,152 17,189 17,36 ! I4,401 14,661 11,729 11,961 9,613
$32,000 27,959 26,154 22,241 21,354 17,297 17,524 14,439 14,783 11,925 12,083 9,698
See footnotes a, b and c on Table 2. d Source: New York State Civil Service Law, Section 1308 131 (1976).
its occurrence, constraints (2) through (12) would have specified minimum values 6. To calculate an individual salary, the individual's relative ratings for the particular characteristics he possesses are multiplied by the factor weights. i
Yi/Xi = S
where S = the individual's salary; Yii = the relative rating of characteristic ] found in factor i which is 6 ',~2ten ?-1 was unrestricted it took on the value of $32,530.
possessed by the individual; Xi = the weight for factor i (given by the solution set). For example, the salary for an Assistant Civil Engineer (14X7), working in Planning (2X1), with supervisory responsibilities of a mini project or "job" (2X2), with an Associates Degree (2Xa), three years of work experience (2X4), one year of experience in title (iXs) and an Intern Engineer's Certificate (2X6) is $15,876. Under the fixed step salary schedule this employee would receive a salary of $14.142. Of course, not all salaries will be greater than that existing under the present schedule. Whether compensation
196
FJ. Fabozzt, A.I¢. Bachner / MP models to determine dvii ~
is more or less depends on the relat~e rating of an individual. If additional resources became available, the linear progranuning model can incorporate the increase into the salary schedule in a logical and consistent manner. Table 4 lists the salary schedules which result from increasin£ the Department's resources of $6,256,058 by 10%, 20%, and 30%. In the generation of alternative salasy structures over time as additional reumrces are made available and the objective function change, it would be possible for an individual salary to decline. The problem is further compounded when new personnel are recruited and present personnel change the characteristics they possess. Such a system of compensation in which one individual's salary declines while a co-worker's salary increases would be totally unacceptable to most employees. However, this problem can be overcome by simply requiring that the optimal weight for each factor in the existing schedule be set as a lower limit when revising the model. This will ensure that no salary will decline. The model optimized the "staff'mg difficulties" factor (Xt) but another factor or salary could also serve as the objective function. For example, the Department might desire to optimize the "supervisory responsibility" factor (,I"2). Optimizing (,!"2) would give employees additional incentive to seek out and accept responsibilities. The optimal weights for each factor in the salary scheme if .I"2 is maximized subject to (1)-(36) is given in Table 2. The optimal salary for each job function is given in Table 3. An example of a salary used as an objective rune-
adarla
tion would be the maximization of the Junior En. gineer's salary. If the Department demed to retain these highly skilled technicians Xs, highest Junior Civil Engineer, should be maximized. Table 2 summarizes these optimal weights and Table 3 provides the salaries for the theoretically high and low for each job function when Xs is maximized. These salaw schedules demonstrate the versatifity of the linear programming model.
S. The goal programming model The deficiency of the linear programming model lies in its inability to assimilate multiple conflicting objectives or priorities. Goal programming, a modification and extension of linear programming, permits such flexibility. For example, if the Department seeks to maximize both the value of the "staff'mg factor" and the salary of the theoretically highest qualified Junior Civil Engineer, the linear program. ming model cannot handle tiffs problem whereas goal programming can. Moreover, if the employees disagree with the objective function set by the Department and seek an alternative objective function, the goal programming model can be used since it can con. sider both objective functions simultaneously. In goal programming each of the conditions (1)-(36) in Table 1 are considered goals and referred to as goal constraints. The objective function of a goal programming model differs from that of linear programming in that it does not contain the decision variables (i.e. factors) in the objective function.
Table 4 Optimal salary schedule with additional resources: XI maximized Prin.
(highest) 0owest) Asso. (highest) (lowest) S.E. (highest) 0owest) A.C.E. (highest) OowesO J.E. (highest) (lowest) S.E.T. (highest) (lowest) Depar~'nent resources
$32,000 27,903 26,153 22,049 21,241 17,205 17,444 14,404 14,737 11,790 12,037 9,640
$35,454 31,136 29,063 24,813 23,890 19,590 19,786 16,283 16,679 13,343 13,910 10,989
$6,256,058
$6,~82,000
a The con~tr~nt that 7,1 be equal :o $32,000 was eli~hlated.
$38,996 a 34,242 31,926 27,256 26,269 21,403 21,617 17,681 18,206 14,565 15,065 11,952 $7,507,000
$41,221 35,679 33,566 28,251 27,262 22,082 22,303 18,262 18,803 15,043 15,611 12,333 $8,133,000
197
F.J. Fabozzi, A.I¢. Bachner / M P models to determine civil service salaries
Instead, negative and/or positive deviational variables are assigned to each goal constraint depending upon whether underachievement or overachievement or both are allowed. Each deviational variable then appears in the objective function according to the priority level assigned to R 7. Goal programming then seeks to minimize the value of the deviational variables. In this application, maximization of the "staff'mg factor" (XI) and the salary of the theoretically highest qualified Junior Civil Engineer Q,s) are assumed. Within a given priority level, different weights can be assigned to a goal s. Tile goal programming model is discussed below. The constraint goals for the theoretically maximum and minimum salary for each job function ( 2 ) - ( 1 2 ) remain as equations in the goal programming model 9. Neither pesitive nor negative deviational variables are introcuced; that is, underachievement or overachievement is not permitted. Similarly, the budgetary goal constraint (36) must be met exactly. The goal constraints ( 1 3 ) - ( 2 8 ) can be rewritten as "greater than cr equal to" goal constraints by simply subtracting the value on the right hand side from both sides of the inequality. Since each goal constraint of the "greater than or equal to" type permits overachievement of a goal, a positive dev. ational variable (d~ for i = 13, 14, ..., 28) is included in each of the respective goal constraints. The goal constraints (29)-(35) which provide monetary bounds for the 7 According to Lee 14, p.27]: "in order to achieve the ordinal solution - that is, to achieve the goals according to their importance - negative and/or positive deviations about the goal must be ranked according to the 'preemptive' priority factors. In this way the low-order goals are considered only after higher-order goals are achieved as desired. If there are goals in k ranks, the 'preemptive' priority factor Pi (] -" 1, 2..... k) should be assigned to the negative and/or positive devlational variables. The 'preemptive' priority factors have the relationship ofP/>>> P]+ 1, which implies that the multip"--_lion of n, however large it may be, cannot make P]+ 1 greater than or equal to Pi"" 8 Within a given priority level, if two or more goals exist, weights must be assigned. Conjoint measurement, a technique developed by mathematical psychologists to quantify judgmental data, could be employed. Using conjoint measures, ranked ordered goals within a preemptive priority level would yield interval-sealed output that can be used as weights. For a primarily expository article on this subject see Green and Rao [ 3 l. 9 The first constraint limited h I tO be less than or equal to $32,000. Hence, this type of constraint will have a negative deviational goal (arT).
factors can be divided Lnto "less than or equal to" constraints for the upper and lower bounds respectively. Negative deviational variables (d/-tj for iU = 29, 30, ..., 35) are assigned to the goal constraints for the upper bounds while positive deviational variables (d~/g for iL = 29, 30, ..., 35) are included in the lower bound goal constraints. Deviational variables which permit the maximization of X1 and Xs must be incorporated into the goal programming model. This is accomplished by the following two goal constraints: (37) XI + d37 = c~ and (38) Xs + d~8 = ~3 where d~7 and d~8 are negative deviational variables and a and/3 are arbitrarily assigned values which are high enough so that they cannot be achieved. The objective function which requires the minlmization of the deviational variables will ensure that X1 and Xs are maximized. The objective function for the goal programming model is then, minimize: ....
SL, d go, d
d3oL, ...,
ou, ..., d 5o, d D
where Pl is tile first (highest) pr~c,rity and 1"2 is tile second (and in tiffs case lowest) priority 10 Two goal programming results are presented in Tables 2 and 3. The first set of optimal val,~es reported is based on the assumption that both goals within the highest priority (i.e. maximization of Xl and X~) are given equal weight. However, as indicated above, different weights can be assigned within a priority level. The second goal programming result is still based on maximizing both XI and Xs as the highest priority, yet in this formulation maximizing Xs is considered to be more important "". The weights assigned to the importance of a variable within a priority level would be set by the Department based on its own objectives assuming the employees do not contest the objectives. In a case where the objectives of the interested parties conflict
10 The computer algorithm to solve this problem is ~ven by Lee [4]. 11 Specifically, the maximization of h s was considered three times more important than the maximization of X 1 for illustration purposes in this formulation.
198
F~I. Fabozzi, A.I¢. Bachner / MP models to determine cidl zervic¢
and both seek that their objective be in the highest priority, then a true compromise position can be attained by using equal weights for both objectives in the highest priority. Depmmres from this weighting scheme depend on the negotiating ability of the interested parties.
6. Conclusion The mathematical approach to salary evaluation permits a consideration of a virtu~y unlimited number of pertinent factors for a job function, the f'mancial resources available to the organization, and the establis~ed objectives of the organization in constructing a salary schedule. The linear programming model of Bruno permits one objective to be considered while the goal programming model allows multiple conflicting objectives. Both models generate salary schedules which overcome the criticisms of the rigid fLxed step salary schedule traditionally used. By parameterizing the bounds for the factor values, the organization can assess alternative salary schedules and determine the necessary budget. Information of this type would be valuable in analyzing the cost of demands made by the bar~at'ning unit representing the civil servants. Moreo';er, the dual solution to the optimal value provide the organization with flexibility :n negotiation since the dual solu. tion indicates the unit costs associated with forcing an incremental doV ,r into the salary weight for each factor. Parameterizing the f'mancial resources avail. able indicates how any increase in funds would be distributed to each factor. This eliminates "across the board" increases that generally accompany the decision to provide additior~al funds. This would not result in a decline in salary for any individual even if
the objective function is changed as long as the optimal monetary value for each factor in the prior period are established as minimum values in subsequent negotiations. There are numerous difficulties that must be surmounted before such models can obtain greater acc¢ptance in helping to determine salary schedules. First, the political power of the bargaining units ou',weigh systems of rationality in issues of selfinterest. Second, many unions seldom encourage or permit significant differentials between members lest their plfilosophy of"all-for-one.and-one-for.alr' be shattered. Finally, Ore programming approach assumes that there are individt~als capable of using such models, patient enough to work out the factors and agree on their relative weights a~,d sincere enough to worry about changing a traditionally accepted form of establishing remuneration for civil servants.
References [ 1] I.E. Bruno, Compensationof school district personnel, Management Sci. 17 (1971) 569-587. [2] A. Chames, W.W.Cooper and R.O. Ferguson,Optimal estimation of executivecompensahon by linear programruing, ManagementSci. 1 (1955) 138-151. [3] P.A. Green and V.R. Rao, Conjoint measurementfor quantifyingjudgmental data, J. MarketingRes. 8 (1971) 355-360. [41 S.M. Lee, Goal Programmingfor Decision Analysis (Auerbach Publishers,Philadelphia,PA, 1972). [5] J. Lennon, Lennor,rebuts radio editorial on merit, Civil Service Leader (February 11, 1977). [6] F.P. Rehmusand H.M. Wagner, Applyinglinear programruing to your pay structure, BusinessHorizons4 (1963) 89-98. [7] J.V. Smith, Merit compensation: The ideal and reality, Personnel J. 51 (May 1972). [8] A. Zack, Meeting the risingcost of public sector settlements, Monthly Labor Rev. 96 (May 1973).