OPTIMIZATION OF RESOURCE ALLOCATION USING INTEGER PROGRAMMING OF IMPROVED RATIO ESTIMATOR UNDER STRATIFIED RANDOM SAMPLING Текст научной статьи по специальности «Естественные и точные науки»

OPTIMIZATION OF RESOURCE ALLOCATION USING INTEGER PROGRAMMING OF IMPROVED RATIO ESTIMATOR UNDER STRATIFIED RANDOM

SAMPLING

Bhatt Ravi Jitendrakumar1, Monika Saini2*, Ashish Kumar3,

Yashpal Singh Raghav4 •

1,2,3Department of Mathematics and Statistics, Manipal University Jaipur,

Jaipur, Rajasthan, India.

4Department of Mathematics, Jazan University, Jazan, Saudi Arabia.

Emails: [email protected], 2*[email protected], 3 [email protected], [email protected]

Abstract

This paper provides a case study that illustrates how integer programming may be used to optimize resource allocation. With the known population median of the study variable acting as auxiliary data, an exponential ratio estimator is shown for estimating the finite population mean under stratified random sampling. The objective is to minimize a cost function within specific bounds. Using integer programming techniques and the Lagrange multiplier approach, we transform the proposed problem into an optimization problem with a linear cost function. This allows us to propose an optimal way for minimizing total costs while maintaining desired accuracy levels. We found that the suggested estimator performed better than methods involving stratified random sampling. Additionally, a numerical example is given to verify the theoretical conclusions for real-world applications. We go over how the problem was formulated, how to use LINGO software to solve it, and the results. It is advised to choose the estimator with the lowest MSE in real-world stratified random sampling situations. The strategy shows significant cost savings and efficient use of resources. The effectiveness of the recommended approach is demonstrated by testing the methodology on both simulated and real-world datasets.

Keywords: linear cost function, integer programming, optimization, resource allocation, lingo software, cost minimization

1. Introduction

The problem of effectively estimating the mean of a study variable in the presence of auxiliary information using different sample procedures has been attempted several times in the literature on sampling theory. The problem of creating effective estimators has been thoroughly researched by a number of authors. Regression estimators, products, and ratios are common examples. Stratified random sampling is the suggested sample design for collecting data from a variety of populations due to its low cost and high efficiency. Allocating resources optimally is essential for increasing productivity and cutting expenses in operations research and management science. Because stratified random sampling can yield estimates that are more accurate than those obtained from plain random sampling, it is a widely used technique in statistical surveys. In order to

maximize estimate precision within budgetary limits, sample sizes must be distributed among different strata. Conventional methods, like Cochran [1] suggested, make use of continuous optimization techniques, which might not be useful when sample sizes have to be integers. In order to determine the best integer solutions for sample size allocation in stratified random sampling, this work investigates the application of Lagrange multipliers and integer programming. Numerous studies have been conducted on the use of simple random sampling [1, 2, 5].

In order to increase estimate precision, a number of scholars have concentrated on maximizing sample size allocation using auxiliary information [2, 5]. Cochran [4] has discussed a number of sampling strategies, including stratified sampling, systematic sampling, simple random sampling, and others. In the topic of survey sampling, Cochran's work is essential since it offers thorough instructions on various methods. In order to increase the efficiency of population parameter estimation, Bahl and Tuteja [6] presents ratio and product-type exponential estimators. Under some circumstances, the suggested techniques perform better in basic random sampling than conventional estimators. The application of optimization theory to large-scale systems is covered in [3], with a focus on computational and mathematical methods for complex system optimization. Neyman [7] contrasted two techniques: purposive selection, which is a non-probabilistic approach, and stratified sampling, which is a probabilistic approach. In order to guarantee representative samples, author suggested stratified sampling. The optimization problem has been expanded to include linear cost functions in more recent research [8,10]. By adding integer restrictions to the optimization issue, this work expands on these foundations and offers a more useful solution for real-world scenarios. Shi et al. [9] examines methods based on optimization, fusing theoretical underpinnings with real-world applications. In order to determine the best integer solutions for sample size allocation in stratified random sampling, [10] investigates the application of Lagrange multipliers and integer programming. In stratified sampling, [11] suggest a technique for calculating the interquartile range under a nonlinear cost function. Their method guarantees accurate and economical estimations for all stratified populations. While the method for creating effective stratum borders in stratified sampling while taking survey expenses into consideration is developed in [12]. The technique lowers the overall cost of the survey while improving sampling efficiency. Recently In stratified sampling, the study [14] suggests the best method for determining the population mean under a linear cost function. Comparing the results to current estimators, they show increased cost-effectiveness and accuracy. In [15], a linear cost function is used to present an efficient and cost-effective estimator for the population mean in stratified sampling. Superior efficiency is demonstrated by the approach, which has been confirmed using real-world data. In order to minimize a cost function under predetermined limits, a resource allocation issue is studied using integer programming techniques. We employ LINGO software to determine the best option and show that this strategy works.

2. Material and Methods

The methodology and optimization strategies employed in this work to create and assess an enhanced median based ratio estimator in stratified random sampling under cost functions are described in this part. The integer programming technique and langrage's multiplier technique were used to solve the optimization issue. Furthermore, the suggested estimator's mathematical characteristics, such as its bias and mean squared error (MSE), are calculated and contrasted with those of other estimators.

I. Study Design

• The study variable (Y) and auxiliary variable (X) are used to split the population into four strata. In order to guarantee that the sample sizes are integer values optimized using integer programming and langrage's multiplier technique, a stratified random sampling design is utilized. The suggested optimization method is validated using the real-world dataset, which is derived from census data. Under the restriction of decreasing the overall survey cost while preserving precision, the ideal sample sizes for each stratum are determined.

Four strata are given in the population, one for each research variable (Y) and auxiliary variable (X).

II. Problem Formulation

• The optimization problem is formulated as follows:

• Minimize the objective function:

4 C-

Minimize ^ — (1)

I=1 ni

Subject to the constraints:

c1 = 2, c2 = 3, c3 = 4, c4 = 5 c0 = 500 2 < nh < Nh h = 1,2,3,4.

The suggested approach was used to ascertain the ideal sample sizes using actual data from [https://censusindia.gov.in/censiis.website/data/censiis-tables]. The findings suggest that when compared to conventional techniques, the integer programming and langrage's methodology produces a more economical use of resources.

3. Solution Techniques

In this instance, a real population from the literature [13] is used to compare the effectiveness of the suggested median-based estimator by [13] with existing estimators. The number of households and the square kilometers of villages and cities, which provide information on study variables and auxiliary variables, respectively, are significant features.

The Neyman allocation is then used to divide the population into four non-crossover strata, and a numerical depiction is finished.

n n NhSh

nh = n—-

Eh=1 NhSh

where i = 1,2, p.

Table 1: Data statistics (source: [13])

Population (N = 645; h = 4)

H Nh nh Yh Mh C2 Cyh c ^-■ymh C2 mh Shh c Jymh \h 8h

1 237 4.13025 116.236 116.81 0.31485 0.20065 0.14554 65.2218 2724.33 0.2379 1.37869

2 164 5.78153 307.603 292.295 0.18397 0.14238 0.30406 131.936 12801 0.16687 0.46825

3 90 16.8718 547.444 548.77 1.64244 2.49501 3.84895 701.592 749552 0.04816 0.64823

4 154 68.2164 757.1 727.165 4.79469 6.20317 8.78042 1657.81 3415068 0.00817 0.70648

Table 2: MSE values of different estimators

Estimators

MSE

¥o(st) Stratified Pp(st) Bahl and Tuteja 1991 Fpe(st) Bahl and Tuteja 1991 ^1(st) Kadilar and Cingi 2004 F2(st) Kadilar and Cingi 2004 ^3(st) Kadilar and Cingi 2004 ^4(st) Kadilar and Cingi 2004 ^5(st) Kadilar and Cingi 2004 P6(st) Kadilar and Cingi 2004 ^7(st) Kadilar and Cingi 2004 p8(st) Kadilar and Cingi 2004 ^9(st) Kadilar and Cingi 2004 Fw(st) Kadilar and Cingi 2004 F11(st) Kadilar and Cingi 2004 P12(st) Kadilar and Cingi 2004

Psubr(st) Subramani 2016 PCR(st) Cochran estimator 1940 p„(st) Yadav 2019

Pprop(st) EStimator

50064.21813 298413.7926 156446.8056 73914.17572 73610.17851 73581.24647 73764.64679 73585.78191 73794.12042 73599.63542 73841.96289 73572.84621 73824.07952 73291.58656 46271.34602 17357.5585 8660.837079 6020.730985 4267.075487

Figure 1: Standard MSE

Table 3: PRE of different estimators

Estimators

PRE

y0(st) Stratified 100

UcR(st) Cochran estimator 1940 578.0529

yp(st) Bahl and Tuteja 1991 16.77678

yp(st) Bahl and Tuteja 1991 32.00079

y1(st) Kadilar and Cingi 2004 67.7329

y2(st) Kadilar and Cingi 2004 68.01263

y3(st) Kadilar and Cingi 2004 68.03937

y4(st) Kadilar and Cingi 2004 67.87021

y5(st) Kadilar and Cingi 2004 68.03518

y6(st) Kadilar and Cingi 2004 67.8431

y7(st) Kadilar and Cingi 2004 68.02237

y8(st) Kadilar and Cingi 2004 67.79914

y9(st) Kadilar and Cingi 2004 68.04714

y10(st) Kadilar and Cingi 2004 67.81557

yn(st) Kadilar and Cingi 2004 68.30827

y12(st) Kadilar and Cingi 2004 108.197

ysubr(st) Subramani 2016 288.4289

y„(st) Yadav 2019 831.5306

yprop(st) Estimator 1173.268

Figure 2: Standard PRE

The comparison of the proposed estimator with existing estimators utilizing stratified random sampling, Tables 2 and 3 unequivocally demonstrate that the proposed estimator has the greatest PRE and the lowest MSE value and their graphs were also given as Figure 1 and 2.

4. Cost Function

The main factor that influences of the number of samples across strata is survey expenditure. [8] introduced linear cost and fixed total cost C0 of the survey as a linear function of n^ h = 1,2,..., L.

L

C0 = £ ChWh (2)

h=1

where Ch denotes the cost per unit of measuring each characteristic in the hth stratum.; h = 1,2,..., L. In this instance, our goal is to determine the fixed linear cost function's least mean square error. Thus, the optimization issue for the proposed estimator in [9] may be described as follows:

Minimize MSE(tpr(st)) L

subject to £ ChWh < C0

h=1

2 < Wh < Nh and Wh are integers; h = 1,2,..., L.

Using the cost function, the mean square error will now be

PP(st)min = Ltf (^f) Cl + ^hCm h - 2hCymh ]. (3)

Integer Programing and Lagrange's Multiplier Technique

Integer Programing:

With a constant linear cost function and actual data, we get the least mean square error. This allows the optimization issue to be stated as follows:

507.3364037 10707.94895 6113.182684 131645.0146 Minimize--\---\---+

Subject to

ni n2 n3 n4

L

£ Chnh < C0 h=1

Bounds on variables:

C1 n1 + C2n2 + C3n3 + C4n4 < C0

2n1 + 3n2 + 4n3 + 5n4 < 500

2 < nh < Nh

and nh are integers; h = 1,2,3,4 2 < n1 < 237, 2 < n2 < 164 2 < n3 < 90, 2 < n4 < 154

The Lagrange multiplier method produces an optimality criterion in some applications. Additionally, the conditions are suitable to set a minimum or maximum. Therefore, the most optimal n value may be found using the Lagrange multiplier method.

The Lagrange function is so defined as:

L(x, A) = f (x) - Ag(x),

where L = Lagrangian, A = Lagrange multiplier, f (x) = Function, x = integer. Now

L(nh, A) = MSE + A ^ L Chnh - C0 j

l=e i^f) a+ftCnh - 2°hcymh \+^ E chnh -. (4)

Now let us partially differentiate the above equation (4) with respect to nh, we get

¿L = 0

dnh

d (el=1 n (f [c2h+cm h - ^Cmh ]+a ( eL=1 chnh - c0))

dnh

Then

0

n ^ - fh)(c2h + dlcmh - ^mh )

nh = v-Ac,-.

Again, differentiate the equation (4) with respect to A, we get dA 0

d (eL=1 n jf [c2h + dh cm h - 2®hcymH ] + A (EL=1 chnh - Q)) ) =

dA

Using the value of equation (4) after differentiating above equation, we get

n- - fh )(cyh + Ql cm h - 2№ymh )ch

-c-' (5)

Now putting the value of equation (5) in equation (4) to find out the value of nh, we get

c0^YZ (1 - fh )(cl + 9h cih - 2^hcymh )

nh

(1 - fh )(c2h + Oh2 cih - 2QhcymU ))c2

c0

nh r • ch

5. Empirical Study with Cost Function

In this part, we prove the efficiency of the proposed estimator using the real data set. The actual population as reported by the Indian census conducted in Lucknow, Uttar Pradesh, is taken into account in the data set (

census-tables). The data N = 645, h = 4, which were used to apply the recommended estimator, contain information on the number of households and the area in square kilometers of certain cities and villages, respectively. These details provide information on the auxiliary variable and the variable under investigation. The population is then split up into four distinct, non-overlapping strata. Integer programming and Lagrange multiplier approaches have been

used in numerical illustration. A reference to the data summary may be found in Table 1. When variables in an optimization problem have to handle integer values, the problem is known as integer programming. If all of the functions are linear, then an integer linear programming problem can be considered. Now, using real data and a fixed linear cost function, we can calculate the least mean square error. Next, the following is a description of the optimization scenario:

Problem Formulation of Proposed Estimator Objective Function

k

^P(sf)min = XX [Cyh + Oh Cmh - 2OhCymh] h=1

Limited population factor will be ignored,

k 1

VP(st)min = XX Yh — [C2h + OhCmh - 2°hCy„h]. h=1 h

The objective is to minimize the cost function defined as:

. . 507.3364037 10707.94895 6113.182684 131645.0146 Minimize--\---\---+

Subject to

ni n2 n3 n4

L

£ chnh < Co h=i

Ci ni + C2n2 + C3 n3 + C4n4 < Co

2ni + 3n2 + 4n3 + 5n4 < 500

Bounds on variables:

2 < nh < Nh

and nh are integers; h = 1,2,3,4 2 < n1 < 237, 2 < n2 < 164 2 < n3 < 90, 2 < n4 < 154

We apply integer programming techniques along with the Lagrange multiplier approach to solve this optimization issue. To determine the best integer values for the sample sizes, the LINGO program is used. Integer variables are used in the model formulation to represent resource allocations, together with an objective function to minimize costs and restrictions to guarantee workable solutions. The variables' ideal values were determined to be ni, n2, n3, and

These numbers show effective resource allocation by minimizing the cost function while meeting all restrictions.

Table 4: Optimized MSE and PRE of different estimators using integer programming

Population (N, h) = (645,4)

Estimators n1 n2 n3 n4 n MSE PRE

P-o(st) Stratified 5 9 37 63 114 37811.71301 100

^p(st) Bahl and Tuteja 1991 4 6 26 74 110 254647.4931 14.84864922

Fpe(st) Bahl and Tuteja 1991 5 6 28 72 111 132124.2009 28.6183097

Pi(st) Kadilar and Cingi 2004 4 6 26 74 110 62734.46491 60.27263173

F2(st) Kadilar and Cingi 2004 4 6 26 74 110 62470.09829 60.52769892

F3(st) Kadilar and Cingi 2004 4 6 26 74 110 62444.95167 60.55207345

F4(st) Kadilar and Cingi 2004 4 6 26 74 110 62606.61364 60.3957167

F5(st) Kadilar and Cingi 2004 4 6 26 74 110 62439.33502 60.55752034

F6(st) Kadilar and Cingi 2004 4 6 26 74 110 62632.11269 60.37112814

F7(st) Kadilar and Cingi 2004 4 6 26 74 110 62460.7517 60.53675624

F8(st) Kadilar and Cingi 2004 4 6 26 74 110 62675.90102 60.32895002

F9(st) Kadilar and Cingi 2004 4 6 26 74 110 62437.71155 60.55909492

Fn(st) Kadilar and Cingi 2004 4 6 26 74 110 62233.04837 60.75825305

Fn(st) Kadilar and Cingi 2004 4 5 28 73 110 38419.51093 98.41799671

Fsubr(st) Subramani 2016 4 16 36 60 116 12234.51456 309.0577304

FCR(st) Cochran estimator 1940 2 3 18 83 106 7361.991722 513.6071112

F**(st) Yadav19 5 20 15 74 114 4412.062967 857.0075561

Fprop(st) Estimator 7 26 17 68 118 2779.875899 1360.194282

Figure 3: Optimized MSE integer programming

Figure 4: Optimized PRE integer programming

In Table 4 the optimized MSE and PRE using integer programing technique is give along with their graphs in Figure 3 and 4.

Table 5: MSE and PRE comparison of different estimators (standard vs integer)

Estimators MSE Optimized MSE PRE Optimized PRE

y0(st) Stratified 50064.21813 37811.71301 100 100

yp(st) Bahl and Tuteja 1991 298413.7926 254647.4931 16.77678 14.84864922

ype(st) Bahl and Tuteja 1991 156446.8056 132124.2009 32.00079 28.6183097

y1(st) Kadilar and Cingi 2004 73914.17572 62734.46491 67.7329 60.27263173

y2(st) Kadilar and Cingi 2004 73610.17851 62470.09829 68.01263 60.52769892

y3(st) Kadilar and Cingi 2004 73581.24647 62444.95167 68.03937 60.55207345

y4(st) Kadilar and Cingi 2004 73764.64679 62606.61364 67.87021 60.3957167

y5(st) Kadilar and Cingi 2004 73585.78191 62439.33502 68.03518 60.55752034

y6(st) Kadilar and Cingi 2004 73794.12042 62632.11269 67.8431 60.37112814

y7(st) Kadilar and Cingi 2004 73599.63542 62460.7517 68.02237 60.53675624

y8(st) Kadilar and Cingi 2004 73841.96289 62675.90102 67.79914 60.32895002

y9(st) Kadilar and Cingi 2004 73572.84621 62437.71155 68.04714 60.55909492

y10(st) Kadilar and Cingi 2004 73824.07952 62653.90433 67.81557 60.35013047

y11(st) Kadilar and Cingi 2004 73291.58656 62233.04837 68.30827 60.75825305

y12(st) Kadilar and Cingi 2004 46271.34602 38419.51093 108.197 98.41799671

ysubr(st) Subramani 2016 17357.5585 12234.51456 288.4289 309.0577304

ycR(st) Cochran estimator 1940 8660.837079 7361.991722 578.0529 513.6071112

y**(st) Yadav19 6020.730985 4412.062967 831.5306 857.0075561

yprop(st) Estimator 4267.075487 2779.875899 1173.268 1360.194282

I Optimized MSE

Figure 5: MSE Comparison (standard vs integer)

Figure 6: PRE Comparison (standard vs integer)

In Table 5 the comparison of MSE and PRE of existing estimator and proposed estimator using integer programming technique is given with their graphs as Figure 5 and 6.

Table 6: Optimized MSE and PRE of different estimators using lagrange's multiplier technique

Population (N, h) -- = (645,4)

Estimators n\ n2 n3 n4 n MSE PRE

P-o(st) Stratified №p(st) Bahl and Tuteja 1991 Fpe(st) Bahl and Tuteja 1991 P-i(st) Kadilar and Cingi 2004 F2(st) Kadilar and Cingi 2004 F3(st) Kadilar and Cingi 2004 F4(st) Kadilar and Cingi 2004 F5(st) Kadilar and Cingi 2004 F6(st) Kadilar and Cingi 2004 F7(st) Kadilar and Cingi 2004 F8(st) Kadilar and Cingi 2004 F9(st) Kadilar and Cingi 2004 Fw(st) Kadilar and Cingi 2004 Fn(st) Kadilar and Cingi 2004 Fn(st) Kadilar and Cingi 2004

Fsubr(st) Subramani 2016 FCR(st) Cochran estimator 1940 2.0 F**(st) Yadav19 5.4

Fpmp(st) Estimator 6.8

4 4 4

4.4 4.4 4.4 4.4

4.3

4.4 4.4 4.4 4.4 4.4 4.3 3.9 4.0

61 6 6

5.7 5.7 5.7 5.7 5.5 5.7 5.7 5.7 5.7 5.7 5.7 5.7 16.6 3.4 20.5 25.3

24

25

27

26.3 26.3 26.3 26.3 26.3 26.3 26.3 26.3 26.3 26.3 26.2 29.2 35.1 18.0 14.6 16.6

43 74 73 73.7

73.7

73.8

73.7

73.9

73.8 73.7

73.7

73.8

73.8

73.9 71.6 60.4 82.8 73.9 68.8

131 110 111 110 110 110 110 110 110 110 110 110 110 110 110 116 106 114 118

37800.05 254555.8

132051

62716.0

62452.1

62427.6 62588.4 62418.1 62614.9

62442.7 62658.1

62420.4

62635.5 62220.1 38373.7 12230.5 7360.1 4410.8

2778.996

100 14.8 28.6

60.3

60.5

60.6

60.4 60.6

60.4

60.5 60.3

60.6 60.3 60.8 98.5 309.1 513.6 857.0 1360.2

300000 250000 200000 150000 100000 50000

MIMMMMi.___

//////////////^my

Figure 7: Optimized MSE lagrange's multiplier

Figure 8: Optimized PRE lagrange's multiplier

Similarly in Table 6 the optimized MSE and PRE using language's multiplier technique is give along with their graphs as Figure 7 and 8.

Table 7: MSE and PRE comparison of different estimators (standard vs lagrange's)

Estimators MSE Optimized MSE PRE Optimized PRE

p0(st) Stratified 50064.21813 37800.05 100 100

pp(st) Bahl and Tuteja 1991 298413.7926 254555.8 16.8 14.8

Fpe(st) Bahl and Tuteja 1991 156446.8056 132050.8 32.0 28.6

pi(st) Kadilar and Cingi 2004 73914.17572 62715.97 67.7 60.3

Pi(st) Kadilar and Cingi 2004 73610.17851 62452.11 68.0 60.5

P3(st) Kadilar and Cingi 2004 73581.24647 62427.61 68.0 60.6

p4(st) Kadilar and Cingi 2004 73764.64679 62588.4 67.9 60.4

P5(st) Kadilar and Cingi 2004 73585.78191 62418.13 68.0 60.6

p6(st) Kadilar and Cingi 2004 73794.12042 62614.93 67.8 60.4

P7(st) Kadilar and Cingi 2004 73599.63542 62442.74 68.0 60.5

p8(st) Kadilar and Cingi 2004 73841.96289 62658.06 67.8 60.3

p9(st) Kadilar and Cingi 2004 73572.84621 62420.37 68.0 60.6

Fw(st) Kadilar and Cingi 2004 73824.07952 62635.47 67.8 60.3

Pn(st) Kadilar and Cingi 2004 73291.58656 62220.05 68.3 60.8

Pn(si) Kadilar and Cingi 2004 46271.34602 38373.67 108.2 98.5

psubr(st) Subramani 2016 17357.5585 12230.45 288.4 309.1

PCR(st) Cochran estimator 1940 8660.837079 7360.138 578.1 513.6

P**(st) Yadav19 6020.730985 4410.764 831.5 857.0

Pprop(st) Estimator 4267.075487 2778.996 1173.3 1360.2

I MSE «Optimized MSE

Figure 9: MSE Comparison (standard vs lagrange's)

Figure 10: PRE Comparison (standard vs lagrange's)

In Table 7 the comparison of MSE and PRE of existing estimator and proposed estimator using langrage's multiplier technique is given with their graphs as Figure 9 and 10.

6. Discussion and Conclusion

In this study, we optimized a new median-based ratio estimator for restricted population means estimation under stratified random sampling. Up to the first level of approximation, bias and MSE formulas are created for the suggested estimators. The suggested estimator was compared theoretically to existing estimators. We determined the conditions in which the suggested

estimator performs better than the traditional estimators. We compare the performance of the proposed estimator quantitatively, considering a real population. The suggested estimator consistently performs better than the existing estimators under stratified random sampling with cost function, both theoretically and numerically. Considering these results, we advise future research to employ the proposed estimator for effective population mean estimation when supplementary data is available. The results indicate a significant reduction in costs through optimal resource allocation. The integer programming and langrage's approach ensures that solutions are both feasible and practical. This methodology can be applied to similar problems in various industries for improved operational efficiency. The problem was successfully solved with the help of LINGO software, which offered a workable solution with either minimizing cost or maximizing precision. To further improve resource allocation tactics, future study might investigate more intricate models and other optimization methodologies.

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