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On the personnel assignment problem
OMEGA, The Int. Jl of Mgrnt Sol., Vol. 1, No. 3, 1973
On the Personnel Assignment Problem GERHARD MENSCH International Institute o f M a n a g e m e n t , Berlin I Received 30 October 1972: revised 22 January 1973)
Recent papers have developed m e t h o d s for personnel assignment under risk o f failure. In this paper, a t w o - p a r a m e t e r model is given. It uses both mean and s t a n d a r d deviation in the stochastic model to bring the risk of failure under control.
THE
DETERMINISTIC
PERSONNEL PROBLEM
ASSIGNMENT
IN THE problem under consideration n open positions, j = 1, 2 , . . . , n, are to be filled. They are such that a candidate i could satisfactorily serve in several positions. There are n candidates, i ~- 1, 2 . . . . . n, and the question is: who shall occupy which position? The problem can be formulated as follows. Let x be an nLvector with variables xij as components, such that 0
(1)
<~ x i j
1
(2)
i~ lx,:i = 1
(3)
2~L~x U ~
Birkhoff [1] pointed out that any linear programming solution to the above system yields values such that x0 ---- 1 if candidate i is assigned to position j, and xo--- 0 if not. This integral property of the set Y ~- [x I (1), (2), (3)} enables large assignment systems to be solved using linear methods. For the optimal assignment, ~, the expected performance, c~, of the group is maximized ('~ =
Max Z ~= l ~ i'~- I coxij,xE Y.
(4)
where cij are individual measures of the expected performances of persons i in positions j. 353
M e n s c h - - O n the Personnel Assignment Problem
O N E - P A R A M E T E R OBJECTIVE F U N C T I O N S Deterministic objective
In one approach [5], the expected performance of person i in position j was evaluated by the positive difference between the ideal "profile of position j " and the "profile of person i", measured by personnel testing procedures [2, 3] in terms of test scores for certain characteristics. With this approach, the expected performance of the group is highest for an assignment ~E Y derived from (4). Probabilistic objectives
King [4] has suggested working with the probabilities, Pu, of success of person i in position j. He outlined a procedure which reduces the scope of the data needed to derive numerical values for the probabilities p~j. The joint probability of success for the group is then I-[(i,j)PijXij, Xe Y
(5)
.
If c~j : logp;j, the linear objective function (4) can be used for the selection of the optimal assignment. Weil [6] has pointed out the weakness of King's criterion (5) in situations where some pij are zero. For several possibly "good" assignments the objective function (5) is then zero. Weil suggested taking the odds of success in the objective function, i.e. the values po/(l-p~j), rather than p~j. The optional assignment, ~, is such as to maximize 1-I(i,j) po..l( l -- P i j ) x i j , x e Y
.
(6)
Weil's criterion (6) fails in those situations in which (5) fails. Furthermore, if any pij = 1, the product (6) is undefined. Weil's criterion can be maximized using (4), with cij : logpu/(l--p~j). Weil [6] found, in a comparison between results derived using linear objective (4) and the odds criterion (6) that the latter "prefers a more equal rather than a less equal division of labour; (0.8, 0.1) is not as good as (0.45, 0.45)". King's criterion (5) has the same equalizing tendency: it avoids assigning persons to jobs for which their individual probability of success Pu or risk of failure (1 --p~j) is high. Well was the first to recognize the need for control of the variability of the performance in the personnel assignment problem.
T W O - P A R A M E T E R STOCHASTIC OBJECTIVES To control the variability of performance, the parameters (cij, trij) of the distributive functionf~j of the random variable cij, the performance of person i in positionj may be used. It is assumed that this random variable cij is normally 354
Omega, Vol. 1, No. 3
distributed. Let ~o denote the expected value of the work performance, e0, with o0 the standard deviation (SD). The joint performance of the group must be evaluated. The expected or average performance of the group is denoted by ~x -----E(cx), and is defined in (4). The variance of the performance of the group is denoted by V(cx), where V(CX) = ~ n= I ~ 7 = 1 oZijx2ij +
covariance, xEY
(7)
where the covariance, denoted by Cov(cx), is Cov(cx) = S ' ] ~ I X j =n I 2'i=l X "s= I xijXrstrOrs, i ~ r, j ~ s
(8)
where crij~sis the bilateral covariance between two persons i and r in the two positions j and s.
L I N E A R A P P R O X I M A T I O N I N TWO S I T U A T I O N S We can now distinguish two cases. Case I: If all individuals' work performance is independent from all others, cr;jrs -----0 and Cov(cx) ---- 0. Expression (8) for the variance simplifies to V(¢X) = ,~Vvn=1 Z~7= l ff2ijX2ij,
x~ Y .
(9)
On the other hand, Case H: If all individuals' work performance is totally correlated with that of all others, the variance becomes
v(~)
=
2'7=1 L'~'=l L'~=l L'"s = l
tTijXijtYrsXrs, XE Y
(10)
and the standard deviation S D ( e x ) = ~/V(cx) is SD(cx) ~ ~'n=l ~'7=10oXiy, xEY .
(11)
To select a personnel assignment which minimizes variability, it is desirable, for computational ease with large scale problems, to use a linear objective function. In Case II the SD(cx) is a linear function. In Case I (9), the integer requirement x0 ~[0, 1] implies that xo = x~. Hence substitution of x0 for x 2 in (9) yields the linear expression V(CX) :
"~=1
n 2 '~'Vj=l tYijXij, x E Y
(12)
J O I N T O P T I M I Z A T I O N W I T H BOTH OBJECTIVES In using criteria (4), (11) or (12), a joint distribution function fi, l---1, 2 . . . . . n!, is selected with mean E(cxt), xicY, and a standard deviation SD(cxt), xtEY. Any solution xlEY selects n individual distribution functions 355
Menseh--On
the Personnel Assignment
Problem
fis with parameters (co, a0). Let xt and x z be two distinct assignments, f~ and f2 the corresponding functions depicted in Fig. 1.
CX
E(cxI) b E(cxz) FIG. I. Comparison of assignments. 0
If the criterion is to minimize the SD(cx) = ~rx or the V ( c x ) ~- cr2x, then the assignment .~ = xl would be selected. This has little variation, but a very inferior expected performance: E ( c x l ) < E ( c x 2 ) . Sub-optimal solutions of this kind can occur also with King's probability criterion [4] and with Weil's odd criterion [6]. A natural way to avoid sub-optimal selections is to use a criterion that takes both objectives into account. That is, we would like to maximize E ( c x ) -~ "cx, and simultaneously minimize V ( c x ) = CF2X or S D ( c x ) = crx, in Cases I or lI respectively. Let a scalar k weigh the relative organizational importance of achieving a high performance level and avoiding higher risk of failure. Then F = c x -- krrZx, x E Y , in Case I
(13)
F = ex -- kcrx, x ~ Y , i n C a s e l l
(14)
are linear parametric programming models, which we shall illustrate by means of an example given in Table 1. TABLE I. DATA TO ASSIGN 5 PERSONS TO 5 POSITIONS
[co] =
21
20
61
17
41
55
28
42
59
-3-7-
17
--455--
13
--~00-- --~--
35
--~33--
t
--19--
[o-iA =
19 --31--
41-- 35--
If we maximize F, k being zero, we maximize cx, xE Y (model (4)). On the other hand, if we maximize F, k being a very large number, the influence of c x is negligible, and the entire weight lies on minimizing ~2x, xE Y (model (12)), or on minimizing ax, x~ Y (model (1 1) ), in either Case I or TI. In this example, 356
Omega, Vol. 1, No. 3
the solutions to (11) and (12) are identical, though this is not generally true. These three extreme cases are given in Table 2. Table 2 also exposes a succession TABLE 2. SOLUTIONS OBTAINED WITH LlNEAR PARAMETRIC MODEL (14)
Parameter k = Points (trx, "~x) =
0
1
2
4
11
--,co
(32,245)
(32,245)
(32,245)
(23,227)
(19,168)
(19,160)
X21 X32 X43 21"54 XI5
X21
X21 -V32 X43 -~'54 X15
X2! X42 X53 X34 X15
XI1
X32 A'43 ~'54 X15
XS1 X42 X33 XI 4 X25
0.5
0.16
0.02
0.00003
-+0
Solutions (x 0 = 1)
Risk
X42 X33 X54 X25
o f points, obtained by parametrical variation of the scaling factor k. This succession o f points, depicted in Fig. 2, constitutes the b o u n d a r y points of the solution set in the plane ~x × crx, using model (11). One o f the b o u n d a r y points is the optimal solution. The decision maker may select that personnel assignment which guarantees a satisfactorily low risk o f failure, preserving a relatively high performance level.
I 250 -
I
I
Max. F.x
1x I 200 -
--
~1
--
I (3z, z45) k =0,1,2
k=4
x
ii
IM
15(~ -
I00
I0
20
2,5
30
SD(cx) =o'x
FIG. 2. Boundary points derived with linear parametric model. REFERENCES I. B1RKHOFFG (1946) Three observations on linear algebra. Riv. Universidad National Tucuman, Ser. A, 5, 147-151. 2. CRONBACH LJ and GLESLER GC (1965) Psychological Tests and Personnel Decisions, 2nd Ed. University of Illinois Press, Urbana, I11. 3. GUION RM (1965) Personnel Testing. McGraw-Hill, New York. 4. KING, WR (1965) A stochastic personnel assignment model. Op. Res., 13, 67-81. 5. MENSCHG (1968) Instrumente der kurzfristigen Personalplanung. Z. Betriebswirtschaft, 38, 469--494. 6. WElL RL (1967) Functional selection for the stochastic assignment model. Op. Res., 15, 1063-1067. 357
On the Personnel Assignment Problem GERHARD MENSCH International Institute o f M a n a g e m e n t , Berlin I Received 30 October 1972: revised 22 January 1973)
Recent papers have developed m e t h o d s for personnel assignment under risk o f failure. In this paper, a t w o - p a r a m e t e r model is given. It uses both mean and s t a n d a r d deviation in the stochastic model to bring the risk of failure under control.
THE
DETERMINISTIC
PERSONNEL PROBLEM
ASSIGNMENT
IN THE problem under consideration n open positions, j = 1, 2 , . . . , n, are to be filled. They are such that a candidate i could satisfactorily serve in several positions. There are n candidates, i ~- 1, 2 . . . . . n, and the question is: who shall occupy which position? The problem can be formulated as follows. Let x be an nLvector with variables xij as components, such that 0
(1)
<~ x i j
1
(2)
i~ lx,:i = 1
(3)
2~L~x U ~
Birkhoff [1] pointed out that any linear programming solution to the above system yields values such that x0 ---- 1 if candidate i is assigned to position j, and xo--- 0 if not. This integral property of the set Y ~- [x I (1), (2), (3)} enables large assignment systems to be solved using linear methods. For the optimal assignment, ~, the expected performance, c~, of the group is maximized ('~ =
Max Z ~= l ~ i'~- I coxij,xE Y.
(4)
where cij are individual measures of the expected performances of persons i in positions j. 353
M e n s c h - - O n the Personnel Assignment Problem
O N E - P A R A M E T E R OBJECTIVE F U N C T I O N S Deterministic objective
In one approach [5], the expected performance of person i in position j was evaluated by the positive difference between the ideal "profile of position j " and the "profile of person i", measured by personnel testing procedures [2, 3] in terms of test scores for certain characteristics. With this approach, the expected performance of the group is highest for an assignment ~E Y derived from (4). Probabilistic objectives
King [4] has suggested working with the probabilities, Pu, of success of person i in position j. He outlined a procedure which reduces the scope of the data needed to derive numerical values for the probabilities p~j. The joint probability of success for the group is then I-[(i,j)PijXij, Xe Y
(5)
.
If c~j : logp;j, the linear objective function (4) can be used for the selection of the optimal assignment. Weil [6] has pointed out the weakness of King's criterion (5) in situations where some pij are zero. For several possibly "good" assignments the objective function (5) is then zero. Weil suggested taking the odds of success in the objective function, i.e. the values po/(l-p~j), rather than p~j. The optional assignment, ~, is such as to maximize 1-I(i,j) po..l( l -- P i j ) x i j , x e Y
.
(6)
Weil's criterion (6) fails in those situations in which (5) fails. Furthermore, if any pij = 1, the product (6) is undefined. Weil's criterion can be maximized using (4), with cij : logpu/(l--p~j). Weil [6] found, in a comparison between results derived using linear objective (4) and the odds criterion (6) that the latter "prefers a more equal rather than a less equal division of labour; (0.8, 0.1) is not as good as (0.45, 0.45)". King's criterion (5) has the same equalizing tendency: it avoids assigning persons to jobs for which their individual probability of success Pu or risk of failure (1 --p~j) is high. Well was the first to recognize the need for control of the variability of the performance in the personnel assignment problem.
T W O - P A R A M E T E R STOCHASTIC OBJECTIVES To control the variability of performance, the parameters (cij, trij) of the distributive functionf~j of the random variable cij, the performance of person i in positionj may be used. It is assumed that this random variable cij is normally 354
Omega, Vol. 1, No. 3
distributed. Let ~o denote the expected value of the work performance, e0, with o0 the standard deviation (SD). The joint performance of the group must be evaluated. The expected or average performance of the group is denoted by ~x -----E(cx), and is defined in (4). The variance of the performance of the group is denoted by V(cx), where V(CX) = ~ n= I ~ 7 = 1 oZijx2ij +
covariance, xEY
(7)
where the covariance, denoted by Cov(cx), is Cov(cx) = S ' ] ~ I X j =n I 2'i=l X "s= I xijXrstrOrs, i ~ r, j ~ s
(8)
where crij~sis the bilateral covariance between two persons i and r in the two positions j and s.
L I N E A R A P P R O X I M A T I O N I N TWO S I T U A T I O N S We can now distinguish two cases. Case I: If all individuals' work performance is independent from all others, cr;jrs -----0 and Cov(cx) ---- 0. Expression (8) for the variance simplifies to V(¢X) = ,~Vvn=1 Z~7= l ff2ijX2ij,
x~ Y .
(9)
On the other hand, Case H: If all individuals' work performance is totally correlated with that of all others, the variance becomes
v(~)
=
2'7=1 L'~'=l L'~=l L'"s = l
tTijXijtYrsXrs, XE Y
(10)
and the standard deviation S D ( e x ) = ~/V(cx) is SD(cx) ~ ~'n=l ~'7=10oXiy, xEY .
(11)
To select a personnel assignment which minimizes variability, it is desirable, for computational ease with large scale problems, to use a linear objective function. In Case II the SD(cx) is a linear function. In Case I (9), the integer requirement x0 ~[0, 1] implies that xo = x~. Hence substitution of x0 for x 2 in (9) yields the linear expression V(CX) :
"~=1
n 2 '~'Vj=l tYijXij, x E Y
(12)
J O I N T O P T I M I Z A T I O N W I T H BOTH OBJECTIVES In using criteria (4), (11) or (12), a joint distribution function fi, l---1, 2 . . . . . n!, is selected with mean E(cxt), xicY, and a standard deviation SD(cxt), xtEY. Any solution xlEY selects n individual distribution functions 355
Menseh--On
the Personnel Assignment
Problem
fis with parameters (co, a0). Let xt and x z be two distinct assignments, f~ and f2 the corresponding functions depicted in Fig. 1.
CX
E(cxI) b E(cxz) FIG. I. Comparison of assignments. 0
If the criterion is to minimize the SD(cx) = ~rx or the V ( c x ) ~- cr2x, then the assignment .~ = xl would be selected. This has little variation, but a very inferior expected performance: E ( c x l ) < E ( c x 2 ) . Sub-optimal solutions of this kind can occur also with King's probability criterion [4] and with Weil's odd criterion [6]. A natural way to avoid sub-optimal selections is to use a criterion that takes both objectives into account. That is, we would like to maximize E ( c x ) -~ "cx, and simultaneously minimize V ( c x ) = CF2X or S D ( c x ) = crx, in Cases I or lI respectively. Let a scalar k weigh the relative organizational importance of achieving a high performance level and avoiding higher risk of failure. Then F = c x -- krrZx, x E Y , in Case I
(13)
F = ex -- kcrx, x ~ Y , i n C a s e l l
(14)
are linear parametric programming models, which we shall illustrate by means of an example given in Table 1. TABLE I. DATA TO ASSIGN 5 PERSONS TO 5 POSITIONS
[co] =
21
20
61
17
41
55
28
42
59
-3-7-
17
--455--
13
--~00-- --~--
35
--~33--
t
--19--
[o-iA =
19 --31--
41-- 35--
If we maximize F, k being zero, we maximize cx, xE Y (model (4)). On the other hand, if we maximize F, k being a very large number, the influence of c x is negligible, and the entire weight lies on minimizing ~2x, xE Y (model (12)), or on minimizing ax, x~ Y (model (1 1) ), in either Case I or TI. In this example, 356
Omega, Vol. 1, No. 3
the solutions to (11) and (12) are identical, though this is not generally true. These three extreme cases are given in Table 2. Table 2 also exposes a succession TABLE 2. SOLUTIONS OBTAINED WITH LlNEAR PARAMETRIC MODEL (14)
Parameter k = Points (trx, "~x) =
0
1
2
4
11
--,co
(32,245)
(32,245)
(32,245)
(23,227)
(19,168)
(19,160)
X21 X32 X43 21"54 XI5
X21
X21 -V32 X43 -~'54 X15
X2! X42 X53 X34 X15
XI1
X32 A'43 ~'54 X15
XS1 X42 X33 XI 4 X25
0.5
0.16
0.02
0.00003
-+0
Solutions (x 0 = 1)
Risk
X42 X33 X54 X25
o f points, obtained by parametrical variation of the scaling factor k. This succession o f points, depicted in Fig. 2, constitutes the b o u n d a r y points of the solution set in the plane ~x × crx, using model (11). One o f the b o u n d a r y points is the optimal solution. The decision maker may select that personnel assignment which guarantees a satisfactorily low risk o f failure, preserving a relatively high performance level.
I 250 -
I
I
Max. F.x
1x I 200 -
--
~1
--
I (3z, z45) k =0,1,2
k=4
x
ii
IM
15(~ -
I00
I0
20
2,5
30
SD(cx) =o'x
FIG. 2. Boundary points derived with linear parametric model. REFERENCES I. B1RKHOFFG (1946) Three observations on linear algebra. Riv. Universidad National Tucuman, Ser. A, 5, 147-151. 2. CRONBACH LJ and GLESLER GC (1965) Psychological Tests and Personnel Decisions, 2nd Ed. University of Illinois Press, Urbana, I11. 3. GUION RM (1965) Personnel Testing. McGraw-Hill, New York. 4. KING, WR (1965) A stochastic personnel assignment model. Op. Res., 13, 67-81. 5. MENSCHG (1968) Instrumente der kurzfristigen Personalplanung. Z. Betriebswirtschaft, 38, 469--494. 6. WElL RL (1967) Functional selection for the stochastic assignment model. Op. Res., 15, 1063-1067. 357