0030-3887/95 $ 2.00 + 0.00 © Operational Research Society of India
Opsearch, Val. 35, No. 4, 1998
Income Estimation : A Goal Programming Model With Rotating Priorities Babu Zachariah Dept. of Quantitative Techniques, Chh. Shahu Central Institute of Business Education & Research, Kolhapur 416 004. Maharashtra, India
Dr. Cherian. P. Kurien Principal, Yashwantrao Chavan Institute of Social Sciences Satara 415001, Maharashtra, India Abstract
Income survey has to carry out the difficult task of estimating the income cf families by measuring different parameters of lite, as people are reluctant to reveal the actual income. The paper discusses a new approach towards the development of such an estimator. Let x1 ' x2 ' ... ,xn be a set of n variables which are linearly related to income (y). Therefore
Collect the sampie observations on these n + 1 variables from m families (m is larger compared to n). Here we use Goal Programming technique to estimate the values of a; 's. Each constraint equation should correspond to a family. 8 i 's would be the decision variables. Hence the information on family can be represented as 8 1 Xi1
+
8 2 X i2
+ ... + 8 n
X
in + d; - d i+
=
Yi
The Goal Programming model minimizes both the deviational variables
i '
t)
(d d on each constraint according to a priority structure. It is best to set a priority structure according' to the reliability cf the information. In this model each family would correspond to a priority level. All possible combinations of priorities are worked out. Select that combination (and the corresponding results) which gives the best correlation between the actual and the estimated income.
MOTIVATION AND SCOPE
This particular technique has evolved from a practical issue wh ich we faced while tackling certain developmental problem. A municipal corporation entrusted us the task of identifying the families below poverty line of implementing the
INCOME ESTIMATION : GLOBAL PROGRAMMING MODEL
347
Nehru Rojgar Yojana(NRY) scheme. In this scheme the selection criterion is the income of families. As people would be reluctant to reveal their actual income, it was decided to estimate the same by using multiple indirect indicators such as expenditure, assets, saving, nature of occupation etc. In order to assign appropriate coefficients of weightage to each of these parameters a new technique was evolved. Also it is important to note that income estimation technique has its application in the following areas. 1.
To assess the developmental resuits in terms of income. In doing so one has to know that the parameters of life may change its weightage tram time to time. Hence it is better to find separate estimators periodically. The proposed technique can be used to this end.
2.
To evaluate the inequalities in income levels between different regions/sections under consideration. This would enable proper planning for any type of developmental activity.
3.
It is weil known fact that the selection of target group is often done with prejudice (certificates from government officials etc.). Income estimation would eliminate such bias to a certain extent.
The income of a family can be expressed in terms of the different parameters attached. It is essential that appropriate weightages are given to these variables. As such there are many techniques which would enable us to do this task, by having information fram a large number of families on different variables. But almost all such techniques assume that all the informants are equally reliable. In reality it is very difficult to find such a situation. Hence it is essential that a technique be developed wherein the more reliable informants would play a better rale in decision making. Goal programming being adecision making model based on a priority structure can serve this purpose. THE GOAL PROGRAMMING MODEL
Goal Programming is adecision making Technique used to tackle mutliobjective allocation problems. Lee[1], Ignazio[2,3,4] and others [5,6,7] have studied the application of this technique to various allocation problems. In this paper the same would be used to decide the weight coefficients in income estimation. Let x 1 ' x2 ' ... , xn be the n variables used to estimate the income denoted as y. Hence y can be written as a linear combination of Xi's Lfl.
348
BABU ZACHARIAH AND DA. CHERIAN P. KURIEN
where a.1 is the coefficient of weightage of x1.. Assurne that information of high reliability is available from m families on these n variables. One may collect these through a pilot survey in the same population where income estimation is to be done. In order to enhance the reliability one mayadopt double sampling and collect m families (from a larger sampie) where there is less disparity between the two sampies on the same family. Also one may employ highly trained personnel to carry out this task. For the purpose of discussion the information thus obtained can be considered the actuals. The coefficients of weightage ought to be estimated from this data which is in the form of an m x n matrix (x .. ] I)
mxn
where x .. is the information on the j I)
th
variable from
the ;th family. Thus we get m constraints. 1.
a1
X 11
+ a2 x 12 + .... + an x1n + d;
d+
=
Y1
2.
a1
X21
+ a2 x22 + .... + an x2n + d;
d+
=
Y2
1
2
....................................
m.
a1 x
m
1
+ a2 xm 2 + .... + an xmn + d- d+ == Y m m m
d;- represents the negative deviation on the i th constraint. d 1.+ represents t he positive deviation on the i th constraint.
Thus we have m constraints representing the income of m families. Sy finding that allocation of a.I 's which would minimize the deviational variables according to a priority structure we will complete the task. As .the informants vary in reliability of information it is always desirable that the highest priority be attached to the most reliable person; the second highest priority to the second most reliable person and so on. But the reliability of an informant is generally unknown. To overcome this difficulty we would adopt the following technique. As there are m constraints with each constraint having a different priority, it is possible that m! combinations of priorities do exist. Out of these m! structures our task is to choose that priority structure which has the optimal reliability attached to it. We may arbitrarily fix a priority structure and run the Goal Program to obtain the values of a.1 's. With this we can estimate the income
INCOME ESTIMATION
GLOBAL PROGRAMMING MODEL
349
of all the m families. Calculate the correlation coefficient between this estimated income and the actual one. Repeat the same procedure for all the m' priority structures. Pick that priority structure and, hence, the values of a.I 's which have given rise to the largest correlation coefficient. Ttle correlation coefficient is calculated using Kari Pearson's formula. Corresponding to every priority structure we obtain the values of ai 's by minimizing simultaneously both the deviational variables. As a second achievement, we can see that the value picked corresponds to the largest correlation coefficient between actual and estimated income. It enables the distribution of the estimated income to reflect the trend found in the distribution of the actual income. As already stated, m > n. However, as the minimization is the simultaneous reduction of both the deviational variables every constraint will reduce the solution space by one dimension. Thus if all constraints are independent, the first n constraints will reduce the solution space in to a point beyond which optimization will not be possible. However, we will include all the m constraints for every run in order to take care of redundant constraints. It is desirable to have the value of m at least as large as n. Otherwise it may deprive us a unique solution. Mathematically no upper limit exists for m. However increasing the value of m beyond a point need not always contribute to the precision of ai's. This is because of the structure of GP. Once the solution space shrinks to a point the latter constraints will be ineffective. MATHEMATICAL FORMULATION AND SOLUTION
Due to practical considerations the model formulation is restricted to three variables on four families. Let
x1 denote number of rooms in the house.
x2 denote expenditure on food per months in rupees. TAßLE 1 : INFORMATION ON THE FOUR SELECTED HOUSEHOLD
Case
X1
)(2
X3
Y
1
2
800
3
60,000
2
5
650
2
36,000
3
1
670
1
24,000
4
3
900
2
54,000
I
BABU ZACHARIAH AND DA. CHERIAN P KURIEN
350 X3
denote number of people employed.
y denote actual family income in rupees.
Let
p. denote i I
th
priority
d + denote positive deviation on i I
th
d ~ denote negative deviation on i I
a. denote weightage of the i I
constraint.
th
constraint.
variable
th
Minimize
such that·
d; - d: = 60000 ........ + d; - d; = 36000
2 a1 + 800 a 2 + 3 a3 + 5 a1
+ 650 a2 + 2 a3
a1 + 670 a 2 + a3 + 3 a1 + 900 a 2 + 2 a3
d; + d~ - d:
d; -
= 24000
=
(A) (B) (C)
54000
(D)
This formulation assumes a priority structure (A,B,C,D). In this notation the constraint with the highest priority( Pl,) is written first. The constraint with the least priority( p 4) is written last. Similarly, we generate 4! priority structures and the results thus obtained on each of the priority structure along with the correlation coefficients between estimated and actual income are given below. The highest value for correlation coefficient is attached with the 10th priority structure. The order of priorities is B,C,A,D with the decision variables a 1 = 0, a2 = 38.18, a3 = 9818.182 . With these values the income of different families is estimated as: case 1
=
Rs. 59999, case 2
=
Rs. 44453
351
INCOME ESTIMATION : GLOBAL PROGRAMMING MODEL
TABlE 2 : FINAL OUTPUT Priority Structure 1 2 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24
a1 0 0 0 7600.001 7600.001 7600.001 0 0 0 0 7600 7600 0 0 0 0 7599.9 7600 0 0 5999.9 5999.9 0 0
a3
a2 0 0 9.917 3.636 3.636 3.636 0 0 9.91 38.18 3.66 3.66 17.39 60 17.39 0 3.636 3.636 17.39 55.38 0 0 17.39 60
20000 20000 17355.37 13963.64 13963.64 13963.64 18000 18000 17355.37 9818.182 13963.64 13963.64 12344.83 0 12347.83 18000 13963.64 13963.64 12347.83 0 18000 18000 12347.83 0
Correlatian Caeft. 0.8895 0.8895 0.9195 0.6032 0.6032 0.6032 0.8895 0.8895 0.9195 0.9954** 0.6033 0.6033 0.9521 0.8098 0.9521 0.8895 0.6032 0.6032 0.9521 0.8098 0.7052 0.7052 0.9521 0.8098
case 3 = Rs. 35399, case 4 = Rs. 53998 There are deviations between estimated and actual income in case2 and case3. Such deviations are possible in any estimation procedure. It should be noted that the same order of magnitude is maintained between actual and estimated income. This could be achieved because the decision making criterion was the best carrelation coefficient; which was achieved at the 10th priority structure. CONCLUSION
One of the coefficients in the obtained solution is zero. Hence we can eliminate this variable in the main survey. When the number of variables in a model is large, correspondingly we have to increase the sam pie size. That means that m! will be a very big number. To solve such a problem a self contained software is required.
352
BABU ZACHARIAH AND DA. CHERIAN P. KURIEN
Other procedures like regression analysis could have been used for estimating ai's. But all such procedures assume that all informants are equaJly reliable and each one will have a say in the decision making. In reality all informants can never be equally reliable; which is the greatest need for using this procedure. One of the drawback of Goal Programming has been considered to be the lack of any quantitative method to fix the priority structure. This modelovercomes that difficulty to some extent. REFERENCES [1)
S.M. LEE. "Goal Programming for Decision Analysis," Philadelphia: Auerback, 1972.
[2]
JAMES P. IGNIZIO. "An Introduction to Goal Programming with Applica~ions to Urban Systems," Computers, Environment and Urban System, Vol. 5, 15-34, 1980.
[3)
...... "Goal Programming A Tool tor Multi objective Analysis," Journal of the Operational Research Society, Vol. 29, 1109-1119, 1978.
[4]
.... "Introduction to Linear Goal Programming," Series: Quantitative Appllcatlons In Soclal Sciences, no. 56, SAGE publications, 1985.
[5]
CHARNES. A AND WW COOPER," Goal Programming and multiple objective Optimizations," European Journal of Operatlonal Research, Vol. 1,307-322, 1977.
[6)
COOK. WD." Goal Programming and Financial Planning Models tor Highway Rehabilitation," Journal of the Operational Research Soclety, Vol. 35, 217, 1985.
[7)
DE KLUYVER. C.A," An exploration of various Goal Programming tormulations - with application to advertising media scheduling, "Journal of the Operatlonal Research Soclety, Vol. 3D, 161-171, 1979.
