Optimum Value of Poverty Measure Using Inverse Optimization Programming Problem

Transcription

1 International Journal of Conteporary Matheatical Sciences Vol. 14, 2019, no. 1, HIKARI Ltd, Optiu Value of Poverty Measure Using Inverse Optiization Prograing Proble Nagwa Mohaed Albehery Departent of Matheatics and Applied Statistics, Faculty of Coerce, Helwan University, Ain Helwan, Cairo, Egypt This article is distributed under the Creative Coons by-nc-nd Attribution License. Copyright 2019 Hikari Ltd. Abstract The proble of poverty is one of the ost iportant probles faced by any countries in the world, especially developing countries. Decision-akers in any countries and international organizations are developing progras and plans to reduce or eliinate the poverty in the countries that suffer fro. In this paper, the new odel will be introduced to iniize the poverty level in a population using the inverse optiization prograing proble. This odel will help the decision aker to deterine the optial distribution of incoe aids that can be given to the poor people in the poorest areas or groups in a population to decrease the poverty. For illustration, our odel will be applied using real data sets collected in Egypt in 2014/2015. Keywords: Poverty easures, poverty lines, distribution of incoe aids, atheatical prograing probles, dual prograing proble, and inverse optiization prograing proble 1. Introduction To alleviate poverty, any progras are developed by decision akers such as increasing the incoe of poor people to decrease the poverty levels in a population. (B. Bidani and M. Ravallion, 1990) introduced nonlinear prograing odel to deterine the optial distribution of incoe aids that give to the poor people to iniize the poverty level and applied the study in Indonesia. (N. M. Albehery, 2003) [2] suggested the deterinistic nonlinear goal prograing odel and the probabilistic nonlinear goal prograing odel, the two odels are introduced to find the optial distribution of incoe aids given to the poor people to iniize the poverty and applied the odels to Egypt data.

2 32 Nagwa Mohaed Albehery Inverse optiization proble is one of the ost iportant probles that are relatively new area of research and have wide applications in industrial areas and econoics. (D. Burton and P. L. Toint, 1992) [5] introduced the inverse optiization. In general, the optiization proble is to find x * ϵ x such that the objective function p(x, c) is optial at x *. The paraeter value of the objective function or the right hand side of the proble can be adjusted as little as possible, so that the given feasible solution becoes optial solution. (J. Zhang and Z. Liu, 1996) [16] introduced soe inverse linear prograing proble. (R. Ahuja and J.B. Orlin, 1998) [1] studied the inverse linear prograing proble under the L and L nor and provided the general for of the inverse optiization for the linear prograing proble. (C. Yang and J. Zhang, 1999) introduced a unifor linear prograing proble to forulate a group of inverse optiization probles and provided two coputation ethods. (C. Heuberger, 2004) [9] presented the various ethods for solving the inverse constraint and unconstrained optiization probles. (N. Mohaed, 2006) [14] presented a coparative study for single and ultiple objective of the inverse optiization probles. (J. Zhang and C. Xu, 2010) [15] provided the inverse optiization for the linearly constrained convex separable probles and constructed the necessary and sufficient conditions to ake the feasible solution is optial solution of the proble. (Y. Jiang, X. Xian, L. Zhang and J. Zhang, 2011) [11] forulated the inverse linear prograing proble as a linear copleentarily constrained iniization proble and proposed the perturbation approach to solve the proble. (S. Jain and N. Arya, 2013) [10] introduced the inverse optiization for transportation probles. (H. A. Mohaed and N. M. Albehery, 2016) [13] introduced the algorith to solve the inverse nonlinear convex probles and presented two applied exaples. In this paper a brief review of poverty easures is presented in section 2. In section 3, atheatical prograing to deterine the optial distribution of incoe aids in the previous studies is presented. In section 4, the definition of the inverse optiization proble is introduced. In section 5, the inverse optiization proble will be used to find the optial distribution of incoe aids that ake the feasible iniu value of poverty easure is optiu. Applied exaple will be presented using real data sets collected in Egypt in 2014/2015 in section 6. The conclusion is presented in section A brief review of poverty easures (J. Y. Duclos and A. Arrar, 2006) [6] defined a poverty easure as the function of rando variable X (incoe or expenditure) and a poverty line z that is the border line between the poor and the non-poor people, denoted by P(x, z), x [0, z) and z (0, ). (J. Foster, J. Greer and E. Thorbock, 1984) [8] introduced the ost popular index in this class is called Foster s index that is defined as: P α (x, z) = 1 n q (z x i i=1 z )α, α 0 (1)

3 Optiu value of poverty easure 33 Where: n is the population size, q is the nuber of poor people in a population, z is the poverty line. When α = 0, the Foster s index is called the headcount ratio, denoted by H. When α = 1, it is called the poverty gap ratio, denoted by P 1 (x, z)when α = 2, it is called the severity of poverty or the distribution of incoe aong poor, denoted by P 2 (x, z). The foster index can be expressed as a weighted su of foster indices in categories or groups of a population as the following: P α (x, z) = ( n j n ) P α (x j, z), α 0, (2) j=1 P α (x j, z) = 1 n j ( z x j α q j i ), α 0 (3) i=1 z Where: is the nuber of categories (areas or groups of people) in a population, n j is a nuber of people in category j, and q j is a nuber of poor people in category j. In this paper, the foster poverty index will be used as the easure of poverty level in a population. 3. Matheatical prograing to deterine the optial distribution of incoe aids (T. Besely and R. Kanbur, 1988) [4] presented a study about the food aids that given to the poorest people in a population as the increasing of their incoe to decrease the poverty levels. The study used the foster index as followed: z 0 P α (x, z) = (1 x+y )α f(x)dx, α 0 (4) z Where y is the aount of incoe aid as the increasing of the poor s incoe. The study derived the change of the poverty easure P α (x, z) when y changes as the following: P α y = α z P α 1, α 0 (5) (N. Kakwani, 1990) [12] calculated the percentage change of poverty easures in each area in a population respect to the incoe ean and the incoe inequality as the following: P i P i μ = η i G Pi + ξ i μ Pi, i = 1, 2,, (6) i G i Where: P i is the poverty easure in area nuber i, i = 1, 2,,, is the nuber of areas in a population, μ i is the ean incoe in area i, is the poverty easure, η Pi is the elasticity of prverty easure respect to the ean incoe in area i, G i is the gini index in area i, ξ Pi, arket price of the fir at risk equals $

4 34 Nagwa Mohaed Albehery 4. The inverse optiization proble 4.1. The atheatical for of prograing proble Consider the following prograing proble: n Min Z = j=1 c j f(x j ), j = 1, 2,, n S.T n j=1 a ij f(x j ) ( = ) b i, i = 1, 2,, x j 0, j = 1, 2,, n (7) Where: n is nuber of decision variables, is nuber of constraints, f(x j ) is the objective function of decision variable x j j, j = 1, 2,, n, c j is the paraeter of decision variables in the objective function, a ij is the paraeter of X j in the constraint i where i = 1, 2,, and j = 1, 2,, n and b j is the constant paraeters in the right hand side of the constraint i, i = 1, 2,, The atheatical for of the dual prograing proble Since the dual prograing proble is iportant to find the solution of the inverse optiization proble, the atheatical for of the dual prograing proble will be defined. The dual prograing proble for the linear prograing proble defined in (7) takes the following for: Max L = i=1 b i f(y i ), i = 1, 2,, S.T i=1 a ij f(y i ) ( = ) c j, j = 1, 2,, n y i 0, i = 1, 2,, (8) Where: is nuber of dual decision variables, n is nuber of constraints, y i is the dual decision variable i, i = 1, 2,,, b i is the paraeter of decision variable in the objective function, a ij is the paraeter of Y i in the constraint j where j = 1, 2,, n and i = 1, 2,, and c j is the constant paraeters in the right hand side of the constraint j where j = 1, 2,, n.

5 Optiu value of poverty easure The atheatical for of the inverse optiization proble If the optial solution for the prograing proble in (7) is (x 0, z 0 ) Where: x 0 and z 0 are the optial value of decision variable and the objective function respectively. Assue there is a feasible possible solution x for the proble in (7) and the decision aker is willing to ake ( x, z 0 ) is the optial solution for the proble. The decision aker wants to ake x is the optial value for the decision variable and keeps the optial value of the objective function z 0 for the proble as the sae value. Then the inverse optiization for the prograing proble in (7) deterinate by adjust the coefficient paraeters c j of the decision variable in the objective function in proble (7). The inverse optiization proble for the prograing proble can be defined by finding the new coefficient d j, j = 1, 2,, n of the objective function to ake ( x, z 0 ) is the optial solution for the proble as follows: Min Z inv = c d p S.T (9) j J a ji f(y j ) d i, i = 1, 2,, j J a ji f(y j ) = d i, i = 1, 2,, y j 0, j = 1, 2,, n Where: y i is the dual decision variable i, i = 1,2,,, J = {j X j = 0}, J = {j X j > 0}, c d p presents the vector nor of degree p. When p = 1, the objective function is linear function but if p 2, the objective function is nonlin ear. To find the solution of the inverse proble defined in (9) when p = 1, the following equation is used to find the optial value of d i : d i = c j c j if c j > 0, X j > 0, i = 1, 2,, c j + c j if c j < 0, X j > 0, j = 1, 2,, n (10) c j O. W, c j = c j j J a ji f(y j ), i = 1, 2,,, j = 1, 2,, n (11) Where: y j is the optial value of the dual decision variable j, j = 1, 2,, n.

6 36 Nagwa Mohaed Albehery 5. The suggested inverse linear prograing proble to find the optial distribution of incoe aids Assue the decision aker is willing to decrease the poverty level in a population by increasing the incoe of the poor people. The decision aker wants to deterine the optial distribution of incoe aids that can be given to the poor people in a population. As we introduced in section 3, (T. Besely and R. Kanbur, 1988) [4] presented a study about the food aids that given to the poorest people in a population by the increasing of their incoe to decrease the poverty levels see equation (4). In this paper, the inverse optiization prograing proble will be used to ake the alternative distribution of incoe aids is the optial distribution and ake the poverty level in this population is iniu value as the following: First: The basic objective function of our odel is iniizing the poverty easure (Foster poverty index) "P α (x, z)" which is defined in (2) by increasing the incoe of poor people in a population as the following: Min P α (x, S, z) = ( n j j=1 P αj (x j, s j, z j ), j=1, 2,,, (12) n ) q j α ) z j P αj (x j, s j, z j ) = (1 x ji+s ji, α 0, (13) i=1 is the nuber of areas or groups in a population, j= 1, 2,,, n is the population size, x j is the ean incoe of poor people in area j, s j is the aount of incoe aid that the decision aker needs to give to area j, z j is the poverty line which is the border line between poor and non-poor people in a population, q j is the nuber of poor people in area or group j.to iniize the poverty easure in (12), soe constraints will be defined as the following: j=1 a ij s j = S, i = 1, 2,, n s j 0 q j, j = 1, 2,, s j = s ji, i=1 j=1,2,, and i=1,2,,qj Where: S is the total aount of incoe aids that is fixed by the decision aker to give to the poor people in a population, s j is the aount of incoe aids that the decision aker needs to deterine to give to the poor people in area or group j, j=1, 2,,, a ij is the paraeter of the decision variable s j, j=1,2,, in constraint i, i=1, 2,, n, this paraeter is related by the characteristics of the poor people in each area or group. a j can be defined as the percentage of poor people, the percentage of the illiteracy, the percentage of uneployent, and the percentage of

7 Optiu value of poverty easure 37 dropout fro education in area j, and n is the nuber of constraints in the proble. Then the prograing odel will be in the following for: Min P α (x, S, z) = ( n j j=1 (1 x ji+s ji n ) q j i=1 α ) z j, α 0, j = 1, 2,, S.T j=1 a ij s j = S, i = 1, 2,, n s j 0, j = 1, 2,, (14) When α 1 the objective function is linear function and when α = 2 the objective function is nonlinear function. Second: Assue (P α, S j ), j = 1,2,, is the optial solution for the odel defined in (14). If the decision aker has a feasible possible solution for the decision variables (S j 0 ), j = 1,2,,. And he wants to ake (P α, S j 0 ), j = 1,2,, is the optial solution for the proble defined in (14) and keeps the iniu value of the optial objective function P α as the sae value. Then the inverse optiization prograing proble defined in (14) will be defined by adjusting the coefficients in the objective function (c j ), j = 1,2,, by detrient the new coefficient paraeter (d j ), j = 1,2,, to ake (P α, S j 0 ), j = 1,2,, is the optial solution for the prograing proble as the following: Min P α inv = c d p S.T (15) j J a ji Y j d i, i = 1, 2,, n j J a ji Y j = d i, i = 1, 2,, n Y j 0, j = 1, 2,, Where: Y j is the optial dual decision variable j, j = 1, 2,,, J = {j s 0 j = 0}, J = {j s 0 j > 0}, c d p is the vector nor of degree p, when p = 1, the objective function is linear function but if p 2, the objective function is nonlinear. Let p = 1, the new value of d i, i = 1,2,, can be calculated by the following for:

8 38 Nagwa Mohaed Albehery d i = c j c j if c j > 0, s j 0 > 0, i = 1,2,, c j = c j c j + c j if c j < 0, s j 0 > 0, j = 1,2,, n c j O. W, j J a ji Y j 6. Applied exaple (16), i = 1, 2,,, j = 1, 2,, n (17) In this paper we will use data fro 2014/2015 that is collected fro the Household incoe, expenditure and consuption survey (HIECS) for the round 2014/2015.This survey was presented by ''Central Agency for Public Mobilization and Statistics (CAPMAS)'' the governent agency to collect data in Egypt. Assue Egypt divided into urban area and rural area, and the decision aker wants to deterine the optial values of the incoe aid will be distributed to the poor people in these areas to decrease the foster poverty easure P α (x, S, z), when α = 1. The household expenditure is used as indicator of welfare to estiate the poverty line. The relative poverty line will be used as the border line between poor and non poor people in a population and it is defined as a percentage of edian of expenditure. (N. M. Albehery and T. Wang, 2011) [3] estiate the lower and the upper liits of the relative poverty line respectively as Z = 1/3 of the edian of annual household expenditure and Z + = 2/3 of the edian of annual household expenditure. The upper liit will be used as the poverty line in this paper and assue the decision aker fix the total aount of incoe aids to distribute to the urban and rural Egypt areas by S =100 illion Egyptian pounds. The following table contains the data used to construct the atheatical odel in (14): Indicators Saple size = n Percentage of poor people Percentage of illiteracy Percentage of dropout fro education Upper liit of relative poverty line = z+ Where: Urban Egypt (1) % 13.5 % 15 % Rural Egypt (2) % 25.5 % 35 % The saple size of people in Urban and Rural Egypt, the percentage of illiteracy, and the percentage of dropout fro education are found fro ''Central Agency for Public Mobilization and Statistics (CAPMAS). 2- The upper liit of relative poverty line z and the percentage of poor people in urban and rural Egypt are calculated using the data fro Household incoe, expenditure and consuption survey (HIECS) for the year 2014/2015 which is presented by Central Agency for Public Mobilization and Statistics (CAPMAS).

9 Optiu value of poverty easure 39 Then the atheatical odel defined in (14) will be the linear prograing odel as the following: Min P 1 (x, S, z) = S S 2 S.T S S 2 = S S 2 = S S 2 = 100 s j 0, j = 1, 2 By solving the above odel using Excel solver, the optial solution will be S 1 = illion EG, S 2 = illion EG and Min P 1 = The dual prograing proble for the above linear prograing proble using the binding constraints can be defined as follow: Max L = 100 Y Y 2 S.T Y Y 2 = Y Y 2 = Y j 0, j = 1, 2 By solving the above dual proble using Excel solver, the optial solution will be Y 1 = , Y 2 = and Max L = Assue the decision aker is willing to ake alternative possible distribution of the incoe aids where the feasible possible aount of incoe aids that are given to urban and rural Egypt by S 1 0 = 2. 95illion EG, S 2 0 = 1. 7 illion EG and he wants to keep the iniu of gap poverty value at P 1 = Then the inverse optiization for the above linear prograing proble will be: Min P 1 = c d inv S.T Y Y 2 = d 1 15 Y Y 2 = d 2 Y j 0, j = 1, 2

10 40 Nagwa Mohaed Albehery d i, i = 1,2 is the optial value of the new coefficient of decision variables in the objective function and Y j, j = 1,2 are the optial dual decision variables can be calculated using equations (16, 17): c 1 = (22. 3( ) ( )) = c 2 = (15( ) + 35( )) = d 1 = d 2 = c 1 + c 1 = = c 2 c 2 = = The new coefficient of the decision variables of the objective function will be ( , ) instead of ( , ) that ake the feasible possible values of the decision variables (2.95, 1.7) are the optial solution for the suggested odel and satisfy the iniu value of the gap poverty easure at Conclusion Inverse optiization prograing proble provides decision akers with an opportunity to take alternative policies. This paper use the inverse optiization proble to ake a feasible possible distribution of incoe aids is the optial distribution to ake the poverty level in a population is iniu value. In our applied exaple, the coefficients of the decision variables in the objective function are adjusted to be new coefficients. The new coefficient ake the possible distribution of incoe aids is optial distribution that iniize the poverty level in a population. Acknowledgeents. I would like to thank Dr. Afaf Aly Hassan El-Dash, professor of operations research at the departent of atheatics and applied statistics and insurance, faculty of coerce and business adinistration, Helwan University for her helpful coents on the initial draft and for her advices to coplete this research. References [1] R. K. Ahuja and J. B. Orlin, Inverse Optiization, Part I: Linear Prograing And General Probles, Working Paper, Indian Institute of Technology, 1998, [2] N. M. Albehery, Probabilistic Prograing to Estiate Poverty Line and Study its Measure, PhD Thesis fro Cairo University, Giza, Egypt, 2003.

11 Optiu value of poverty easure 41 [3] N. M. Albehery and T. Wong, Statistical Inference of Poverty Measures Using U-Statistics Approach, International Journal of Intelligent Technologies and Applied Statistics, 2 (2011), [4] T. Besley and R. Kanbur, Food subsidies and poverty alleviation, The Econoic Journal, 98 (1998), [5] D. Burton and P. L. Toint, On an instance of the inverse shortest paths proble, Matheatical Prograing, 53 (1992), [6] J. Duclos and A. Arrar, Poverty and Equity: Measureent, Policy and estiation with DAD, Springer Science & Business Media, New York, USA, [7] Egyptian Central Agency of Statistics, Incoe, Consuption and Expenditure Research (2014/2015), Egyptian Central Agency of Statistics, Cairo, Egypt, (2015). [8] J. E. Foster and E. A. Thorbecke, A class of decoposable poverty easures, Econoetrica, 52 (1984), [9] C. Heuberger, Inverse cobinatorial optiization: A survey on proble, Methods and Results, Journal of Cobinatorial Optiization, 8 (2004), [10] S. Jain and N. Arya, An Inverse Capacitated Transportation Proble, IOSR Journal of Matheatics, 5 (2013), [11] Y. Jiang, X. Xiao, L. Zhang and J. Zhang, A perturbation approach for a type of inverse linear prograing probles, International J. of Coputer Matheatics, 88 (2011), [12] N. Kakwani, Poverty and Econoic Growth, World Bank, Washington. D. C, USA, [13] H. A. Mohaed and N. M. Albehery, Applications of Inverse Optiization for Convex Nonlinear Probles, Institute of Statistical Studies and Research, Cairo University, 2016, [14] N. Mohaed, A Coparative Study of the Inverse Optiization Probles, PhD Thesis, Ain Shas University, 2006.

12 42 Nagwa Mohaed Albehery [15] J. Zhang and C. Xu, Inverse optiization for linearly constrained convex separable prograing probles, European Journal of Operational Research, 200 (2010), [16] J. Zhang and Z. Liu, Calculating soe inverse linear prograing probles, Journal of Coputational and Applied Matheatics, 72 (1996), Received: February 15, 2019; Published: March 15, 2019