A mathematical programming approach to salary administration

A mathematical programming approach to salary administration

Comput & Indus.Engng Vol. 7, No. I,pp, 7-13, 1983 Printed in Great Britain. 0360-83521831010007--07503,00/0 Pergamon Press Ltd. A MATHEMATICAL PROGR...

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Comput & Indus.Engng Vol. 7, No. I,pp, 7-13, 1983 Printed in Great Britain.

0360-83521831010007--07503,00/0 Pergamon Press Ltd.

A MATHEMATICAL PROGRAMMING APPROACH TO SALARY ADMINISTRATION ALBERTOGARCIA-DIAZand GARYL. HOGG Departmentof Industrial Engineering,TexasA&M University,CollegeStation,TX 77843,U.S.A. (Recdved in revisedform November 1981)

Abstract--This paper discusses a mathematicalprogrammingapproach to designingan executivesalary administrationguidebased on the importanceof the positionsand the qualityof job performance The two fundamentalconceptsof the salary systemthat is consideredin this studyare the compensationplan and the classificationplan. The purposeof the compensationplan, and the objectiveof the paper, is to develop a salaryguidefor givenresults from the classificationplan. The paper assumes an understandingof basic principlesof job evaluationand someexposureto formulationof mathematicalmodels.Emphasisis put on the formulationand linearizationof the models, rather than on developmentof solution techniquesthat would exploitthe particularstructureof the models. This paper develops a quantitative approach to the design of a salary administration guide for executive personnel. The guide is defined as a set of decisions to control the value and timing of salary increases in terms of both the performance of individual employees, and the relative importance of their positions. Other salary actions such as general increases, promotional increases, and reclassification increases will not be considered. To increase the reliability and flexibility of the proposed methodology, such concepts as those of potential for advancement and position in the salary range will be used. The paper consists of three sections. Section 1 presents basic definitions of salary administration relevant to the problem to be considered, and a summary of related models. Section 2 contains the description, formulation, and linearization of the proposed models. Section 3 summarizes the overall approach and the worst-case results of the computational analysis of the model. 1. GENERAL BACKGROUND The following definitions are essential to the understanding of the basic principles of job evaluation, and, therefore, to the correct interpretation of this paper: Merit Increase: a salary increase recommended or granted in recognition of consistently satisfactory job performance as measured in terms of quality and/or quantity. Position Group: group where a position is classified according to the scope of the functions and responsibilities involved. Performance Category: group where an individual employee is classified according to the quality of the employee's job performance. Salary Range: the extent of salaries, from low to high, paid by the company to employees filling out positions classified in a given group. Range Position: the amount by which a given salary exceeds the salary range minimum, expressed as a percentage of the range. Interval: time elapsed between two successive merit increase actions concerning a given individual employee. Potential: ability of an individual employee to advance in the organization. The design of executive compensation plans is a complex problem. Although a significant number of books and papers address this topic, the availability of sound mathematical models is currently limited. A pioneering linear programming approach due to Charnes, Cooper, and Ferguson[2] finds weights for the factors included in a compensation plan, such that external consistency is maximized without violating internal consistency. A similar methodology, developed by Rehmus and Wagner[5], uses linear programming to find a weighting system that minimizes the sum of discrepancies between factor ratings and current salaries for all the jobs being analyzed. Bruno [1] has developed a linear programming salary evaluation model to compensate school district personnel without violating hierarchial and budget conditions. A contribution by Fabozzi & Bachner[3] describes how linear programming and goal programming can be applied to determine a civil service salary structure in terms of service grade and seniority.


Recently, Handelman[4] developed a model of a salary raise program in terms of a mechanism which links future percentage increases to professional age; given a starting mean salary distribution, salaries can be forecast for a specified period.

2. MATHEMATICALMODELS Sound company policies require the preparation of an annual merit increase budget to provide the funds needed to cover all the adjustments in the compensation of employees deserving merit recognition. The use of a salary guide is of great value in the preparation of this budget, because it allows for the planning of both the value and the timing of salary adjustments. Two mathematical models are proposed for developing a salary guide. The first model yields the salary increases. The second model, using the results of the first model, gives the timing of those adjustments, thus allowing for the determination of intervals between increases for each employee.

2.1 The salary increase model The proposed model for determining salary increase percentages, referred to as Model I, takes into consideration fundamental characteristics of the specific group of personnel under consideration, such as current salaries, performance grades, position groups, salary ranges, and potentials for advancement. 2.1.1 Formulation of Model L The following notation is used in the formulation. The symbols used are either input parameters or variables. For the various subscripts, the corresponding ranges are also given. Input Parameters N number of executive employees eligible for merit increases M number of increase categories in the salary guide ei current evaluation of employee i's performance. It is assumed that ei takes on only one of the values 1, 2. . . . . M. The highest performance grade is M and lowest is 1. E set of employees with low potential for advancement C~ current salary of employee i R~ prevailing salary for the position of employee i Ui upper bound on the salary of position of employee i Li lower bound on the salary of position of employee i a~ lower bound on salary increase percentages (0-< a~ -< !) a2 upper bound on salary increase percentages (a~ -< a2--- 1) a3 minimal difference between increase percentages of neighboring categories (0--< a3-< a2) d maximal deviation, in increase categories, allowed between job performances and salary treatments q maximal position in salary range allowed to low-potential personnel (0 <- q <- 1) Variables Xm increase percentage corresponding to category m Zim = l, if employee i is given salary treatment m; Zim = 0, otherwise Ranges of Subscripts i =1,2 . . . . . N m =1,2 . . . . . M The formulation of the model follows: Minimize


A mathematicalprogrammingapproachto salaryadministration


Subject to

Ci(~m ZimXm+ l) < Ui, for all i~ E Ci(~ZimX,,+ Ci(~Zi,,X,,+

1 ) -<

1) >- Li, for all/

Li+ q(Ui- Li), for all

Xl~al, XM<-a2, Xm-Xm_t>-a3, E



iE E


for all m ~ 2


mZim <- el, for all i




mZim >- ei-

d, for all i



mZi,, =

1, for all i



Z / r a = 0,

1, for all i and all m.


Note that the objective function (1) and constraints (2), (3) and (4) are not linear. In these expressions, Ci(Y~ZimXm+ 1) is the new salary of employee i. The objective function defined in m

(1) minimizes the overall cumulative deviation between the new salaries and the prevailing ones, hence maximizing the level of "external consistency" of the salary administration guide. Constraints (2) and (3) represent the condition that individual salaries should be within given ranges dependent upon the importance of the positions. This condition is known as "internal consistency". Constraint (2) defines the upper bounds on new salaries, and constraint (3) establishes the lower bounds. Constraint (4) imposes an additional limit on the salaries of tow-potential employees in order to allow for future (and perhaps modest) increases, even when the prevailing salary tendency changes very little. Constraints (5) indicate that merit increases must not exceed a given limit, and that must not decrease as the quality of job performance increases. It is assumed that the number of performance categories is equal to the number of salary increase categories. For neighboring increase categories, the difference between the corresponding percentages must be attractive enough to motivate the personnel to improve job performances. According to constraints (6) and (7), no employee should be assigned to a salary increase category higher than the employee's job performance grade. Constraint (6) established a maximal increase category, and constraint (7) defines a range for salary treatments. That is, a salary treatment may lie below the job performance grade but must not deviate from it by more than a specified number of categories. This condition adds flexibility to the salary guide. It allows for some degrees of freedom in classifying individuals whose new salaries would go beyond the salary limits of their positions if the salary treatments were equal to the job performance grades. Finally, constraints (8) and (9) specify that each eligible employee must be assigned to exactly one salary increase category. As a guideline, it is usually acceptable to estimate the number of recommended merit salary increases between 35% and 40% of the total number of positions, in a given annual budget. In the model under consideration, the number of effective variables and constraints can be reduced, sometimes significantly, after predetermining some of the Z,-,, variables to be equal to



0 or l, as follows: (a) in constraint (6), Z/m = 0 when m >- ei + 1 (b) in constraint (7), Zim = 0 when m -< ei - d - 1, and (c) Zil = 1 and Zim =0, for m-->2, when ei = 1. After Zim is set equal to the predetermined value indicated in (a), constraint (6) becomes redundant, and hence can be removed from the model. Similarly, after Zim is set equal to the predetermined value indicated in (b), constraint (7) becomes redundant and can be deleted from the model. Also, constraint (8) is not necessary for employees whose Z/m has been predetermined to be equal to 1 by (c) above. Constraint (3) need not be specified when it is already satisfied by the current salary of a given employee. Also, employees whose current salaries violate constraint (2) must not be included in the analysis (those employees are usually referred to as "red circles"). 2.1.2 Linearization of Model L The objective function of (1) and constraints (2), (3), and (4) can be linearized as follows. The appropriate linear objective function to be minimized is a sum of non-negative variables ti and hi, as shown in reference [6]. In the conversion, N constraints are added:

C~(~mZimXm+ 1 ) - R i = h i - 6 , for all/


C~(~mZ~,,Xm+ 1 ) - h ; + ti = Ri, for all/


ti -> 0 and hi >- O, for all i


or, equivalently,


The resulting linear objective function in the transformed model is given by Minimize ~,(hi + ti) i


In order to linearize constraints (2), (3), (4) and (11) define the transformation

W~m =Zi.X.,


Note that

Xm if Z#.= l (15)

W,r, =

Oif Zi., = O. The following additional constraints guarantee that relation (15) is satisfied: IV;,. _< Zim


Wi. >__Zi,. + X m - 1


Wim ~Xm.


The transformed model is a mixed integer linear program, usually referred to as an MILP model. The MILP model which is equivalent to Model I minimizes the objective function given

A mathematical programming approach to salary administration


in (13) subject to constraints (2), (3), (4), (5), (8), (9), (11), (12), (16), (17) and (18). In constraints (2), (3), (4) and (11) use is made of the transformation defined in (14). It is assumed that the linearization of Model I is performed after the reduction of the number of variables and constraints. If Z , is predetermined to be equal 1, then Z~m = 0 for all m -> 2. Equivalently, W, = Xl and W~,, = 0, for m -> 2. In this case, constraints (8), (16), (17) and (18) are not needed for employee i. If Zi,, = 0, then W/m = 0, and constraints (16), (17) and (18) are not needed for the corresponding combination of values of i and m. 2.2 The interval assignment model This model will be referred to as Model II. The solution to Model I provides values Z*m which can be used to determine the salary treatment of employee i, as indicated by g~ = EmZ*m. ol

The interval corresponding to employee i is evaluated by Model II and depends on the value gi derived from Model I. 2.2.1 Formulation of Model II.

Input Parameters A B

lower bound on intervals upper bound on intervals (B > A) bt ideal number of salary adjustments desired in period t f~ number of periods between the last increase of the salary of employee i and the beginning of the planning horizon. It is assumed that A -> [~ + 1 g~ salary treatment given to employee i by Model I, gi = EmZim m

T number of periods in the planning horizon /3i = min { T; B fi} -

Variables Y,t

= 1, if employee i gets an increase in period t; Y~t =0, otherwise. Here, i = 1,2 . . . . . N a n d t = l , 2 . . . . . T

Model Minimize

Y~t - bt



Subject to T

~. Y,t = 1



~, tY~,+fi = ~, tY~t+fj, for i and j such that gi =gi

t=A fi





~, tYit+fi <- ~, tYit+f s, for i and j such that gi>gj







tYit +fi <- B, for i corresponding to minimal gi


tYit +fi >- A, for i corresponding to maximal gi




Y/t = 0, 1, for all i and all t


The effective size of this model can also be reduced by noting that (a) for t and i such that



t < A - f i or t > fli the value of Y,'t can be predetermined to be equal to 0, and (b) for each period such that Y~t is predetermined to be equal to 0 for i = 1, 2 . . . . . N, the corresponding term can be removed from the objective function defined in (19). The purpose of the objective function (19) is to schedule the salary adjustments in such a way as to maximize the goodness of fit of the scheduled increases to a desired ideal distribution along the planning horizon for which the salary guide is to be designed. This desired distribution stipulates an ideal number of salary adjustments for each period of the horizon. For example, a company may be interested in a distribution of increases as uniform as possible; or another company may want as few adjustments as possible in some special periods of the horizon. Both of these policies have practical advantages from a managerial point of view. According to constraint (20), each salary increase is scheduled exactly once in the specified planning horizon. Constraints (21) and (22) insure that if any two employees belong to the same increase category, their intervals must be equal, but if one of the two employees belongs to a higher category, that employee's interval must not be longer than the other employee's. This condition expresses the policy that better job performances are recognized with shorter intervals. Constraint (23) places an upper bound on the interval corresponding to any employee in the group with the lowest salary increase classification, and, with constraints (21) and (22), extends the upper bound condition to all other employees. A similar argument holds in the case of lower bounds on intervals, stipulated by constraint (24). 2.2.2 Linearization o f M o d e l II. The procedure followed in this section is identical to that for the linearization of the objective function of Model I. It is also assumed that Model II has been reduced to a minimal size by predetermining some of the Yit's to take on the value zero~ as explained in Section 2.2.1. Essentially, the linearization is achieved by replacing the objective function defined in (19) with a new one given in (26) and incorporating the constraints (27) and (28), where 7t and ~ are non-negative variables:

Minimize ~(3't + 6,)



~ , Yit + ~t - 7t = bt,

t = 1. . . . .






t=l ..... T


3. CONCLUDING REMARKS In this paper the salary administration of executive personnel has been discussed in terms of two MILP models which allow for the development of a salary treatment guide that includes both increase percentages and intervals as functions of employee job performances. The proposed methodology maximizes external consistency subject to given conditions on internal consistency. Table 1 summarizes the upper bounds on the number of variables and constraints for each model. Ordinarily, the actual number of constraints and/or variables is substantially smaller than the given bound.

Table 1. Upperboundson numberof constraintsand variables Integer














A mathematical programming approach to salary administration


REFERENCES I. J. E. Bruno, Compensation of school district personnel. Management Sci. 17, 56%587 (1971). 2. A. Charnes, W. W. Cooper & R. O. Ferguson, Optimal estimation of executive compensation by linear programming. Management Sci. 1, 138-151 (1955). 3. F. J. Fabozzi & A. W. Bachner, Mathematical programming models to determine civil service salaries. Europ. J. Oper. Res. 3, 190-198 (1979). 4. G. H. Handelman, Salary raises by percent increases. Decision Sci. 12, 322-337 (1981). 5. F. P. Rehmus & H. M. Wagner, Applying linear programming to your pay structure. Business Horizons, Winter 1963, pp. 89-98. 6. H. M. Wagner, Principles of Operations Research. Prentice-Hall, Englewood Cliffs, New Jersey (1969).