Integer programming formulations applied to optimal allocation in stratified sampling 4. Numerical resultsInteger programming formulations applied to optimal allocation in stratified sampling 4. Numerical results

This section provides results for the application of the selected multivariate optimum allocation approaches to a set of population datasets. The approaches considered include:

The BSSM algorithms developed to solve Formulations C and D provided in Section 3;
An improved version of Bethel’s algorithm (Bethel 1989) developed by Ballin and Barcarolli (2008);
The textbook method proposed in Cochran (1977, Section 5.A.4).

Eleven population datasets were used for the numerical illustration, but for space considerations, here we report only the results for three of these populations. The three selected populations are described in tables A1 through A6 in Appendix A. Table A1 provides a brief description of each survey population and provides the list of the corresponding survey variables. Table A2 provides information about how each population was stratified prior to determining the optimum allocation. In particular, for the survey dataset called $M u n i c S w$ the strata had been previously defined. The other two populations were stratified using a stratification algorithm available in the R package $s t r a t i f i c a t i o n$ or a classic $k - means$ clustering method available in the $b a s e$ R package.

Table A3 presents the number of population strata $(H),$ the number of survey variables $(m),$ and the population size $(N)$ for each of the populations considered. Tables A4 through A6 provide the population counts, means, and standard deviations per stratum for the survey variables considered in each of the three survey populations considered.

The results of all the numerical experiments reported here were obtained using the R packages and functions mentioned, and using a Windows 7 desktop computer with 24GB of RAM and with eight i7 processors of 3.40GHz. Processing time ranged from miliseconds (for the relatively small $M u n i c S w$ population) to less than 4 seconds (for the larger $S c h o o l s N o r t h e a s t$ population, under formulation C). This demonstrates that the proposed formulations provide a feasible and efficient alternative for multivariate optimum allocation problems of small and medium size, for populations of sizes $(N)$ in thousands and even tens of thousands.

Tables 4.1 through 4.3 provide the target coefficients of variation $({CV}_{j})$ for each of the survey variables, the sample sizes obtained using the algorithm to solve proposed Formulation C $(n_{BSSM})$ and Bethel’s algorithm $(n_{Bethel}),$ and the achieved coefficients of variation for the estimators of totals of the survey variables considered in each population under the two algorithms compared.

Table 4.1
Results for the $C o f f e e F a r m s$ population
Table summary
This table displays the results of Results for the $C o f f e e F a r m s$ population. The information is grouped by XXXX
XXXX (appearing as row headers), Algorithm for Formulation C and Bethel’s Algorithm, calculated using XXXX and XXXX
XXXX units of measure (appearing as column headers).
${CV}_{j}$ $(%)$	Algorithm for Formulation C				Bethel’s Algorithm
${CV}_{j}$ $(%)$	$n_{BSSM}$	$CV (t_{1})$ $(%)$	$CV (t_{2})$ $(%)$	$CV (t_{3})$ $(%)$	$n_{Bethel}$	$CV (t_{1})$ $(%)$	$CV (t_{2})$ $(%)$	$CV (t_{3})$ $(%)$
5	2,545	1.24	5.00	2.92	2,546	1.23	5.00	2.91
10	754	3.30	10.00	7.01	755	3.30	9.99	7.07
15	347	5.21	15.00	11.01	349	5.11	14.95	10.85

Table 4.2
Results for the $S c h o o l s N o r t h e a s t$ population
Table summary
This table displays the results of Results for the $S c h o o l s N o r t h e a s t$ population. The information is grouped by XXXX
XXXX (appearing as row headers), Algorithm for Formulation C and Bethel’s Algorithm, calculated using XXXX and XXXX
XXXX units of measure (appearing as column headers).
${CV}_{j}$ $(%)$	Algorithm for Formulation C			Bethel’s Algorithm
${CV}_{j}$ $(%)$	$n_{BSSM}$	$CV (t_{1})$ $(%)$	$CV (t_{2})$ $(%)$	$n_{Bethel}$	$CV (t_{1})$ $(%)$	$CV (t_{2})$ $(%)$
2	1,624	2.00	1.79	1,628	2.00	1.78
5	294	5.00	4.31	299	4.96	4.23
10	80	9.93	8.24	83	9.72	8.13

Table 4.3
Results for the $M u n i c S w$ population
Table summary
This table displays the results of Results for the $M u n i c S w$ population. The information is grouped by XXXX
XXXX (appearing as row headers), Algorithm for Formulation C and Bethel’s Algorithm, calculated using XXXX and XXXX
XXXX units of measure (appearing as column headers).
${CV}_{j}$ $(%)$	Algorithm for Formulation C					Bethel’s Algorithm
${CV}_{j}$ $(%)$	$n_{BSSM}$	$CV (t_{1})$ $(%)$	$CV (t_{2})$ $(%)$	$CV (t_{3})$ $(%)$	$CV (t_{4})$ $(%)$	$n_{Bethel}$	$CV (t_{1})$ $(%)$	$CV (t_{2})$ $(%)$	$CV (t_{3})$ $(%)$	$CV (t_{4})$ $(%)$
5	1,527	2.01	3.88	5.00	4.41	1,529	2.00	3.88	4.99	4.40
10	761	3.61	7.27	9.99	8.77	763	3.60	7.25	9.97	8.75
15	439	5.01	10.22	14.98	13.07	441	4.95	10.16	14.94	13.03

As expected, in all cases the sample sizes obtained by solving Formulation C were smaller than (bold) or equal to those obtained using Bethel’s algorithm. However, the improvements were generally not substantial. Nevertheless the proposed algorithm managed to improve upon the current best method in the nine scenarios considered (three populations times three levels for the target CVs). The improvements appeared to be a bit larger for the $S c h o o l s N o r t h e a s t$ Population, where the number of strata is also larger. Similar results (not shown here for conciseness but available from the authors on request) were obtained for the other eight populations considered in an initial version of the paper.

Tables 4.4 to 4.6 provide the results of applying Formulation D and the textbook method proposed in Cochran (1977, Section 5.A.4) to the same three survey populations. Now the goal is to minimize the weighted relative variance of the HT estimates of total, while keeping the overall sample size or cost. The first line in each of these tables contains the total sample sizes considered for the allocation. These sample sizes correspond to sampling fractions of 10%, 20% and 30% of the corresponding population sizes $(N)$ respectively, as indicated in the second line in each of the tables. The subsequent lines provide the allocation of the total sample into the strata, the coefficients of variation achieved for the HT estimates of totals of the survey variables considering the allocation, and the sum of the coefficients of variation $(Σ CV (t_{i})),$ which is a summary measure of efficiency across all survey variables.

The importance weights were taken as equal across all survey variables, and the unit survey costs were taken as equal across all strata, in each population, for these applications.

Table 4.4
Results for the $C o f f e e F a r m s$ population
Table summary
This table displays the results of Results for the $C o f f e e F a r m s$ population. The information is grouped by XXXX
XXXX
XXXX (appearing as row headers), 2.047
10%, 4.094
20% and 6.142
30%, calculated using BSSM-D and Textbook units of measure (appearing as column headers).
$n$ $S a m p l i n g f r a c t i o n$ $R e s u l t$	2.047 10%		4.094 20%		6.142 30%
$n$ $S a m p l i n g f r a c t i o n$ $R e s u l t$	BSSM-D	Textbook	BSSM-D	Textbook	BSSM-D	Textbook
$n_{1}$	1,174	1,124	2,483	2,340	3,792	3,625
$n_{2}$	662	737	1,400	1,544	2,139	2,306
$n_{3}$	211	186	211	210	211	211
$CV (t_{1})$	1.02	1.14	0.62	0.62	0.42	0.42
$CV (t_{2})$	5.78	5.79	3.62	3.65	2.61	2.63
$CV (t_{3})$	2.86	2.98	1.73	1.73	1.19	1.17
$Σ CV (t_{i})$	9.66	9.91	5.97	6.00	4.22	4.22

Table 4.5
Results for the $S c h o o l s N o r t h e a s t$ population
Table summary
This table displays the results of Results for the $S c h o o l s N o r t h e a s t$ population. The information is grouped by XXXX
XXXX
XXXX (appearing as row headers), 7,508
10%, 15,017
20% and 22,525
30%, calculated using BSSM-D and Textbook units of measure (appearing as column headers).
$n$ $S a m p l i n g f r a c t i o n$ $R e s u l t$	7,508 10%		15,017 20%		22,525 30%
$n$ $S a m p l i n g f r a c t i o n$ $R e s u l t$	BSSM-D	Textbook	BSSM-D	Textbook	BSSM-D	Textbook
$n_{1}$	82	58	82	60	82	66
$n_{2}$	36	33	62	53	53	62
$n_{3}$	7	6	7	6	7	6
$n_{4}$	206	214	465	433	771	611
$n_{5}$	1,083	1,000	2,091	1,962	2,671	2,121
$n_{6}$	447	452	891	914	1,428	1,436
$n_{7}$	361	371	711	750	1,182	1,175
$n_{8}$	2,995	2,989	5,963	6,055	9,088	9,634
$n_{9}$	976	1,023	1,965	2,069	3,078	3,229
$n_{10}$	399	419	800	849	1,331	1,338
$n_{11}$	797	813	1,742	1,647	2,596	2,612
$n_{12}$	119	130	238	219	238	235
$CV (t_{1})$	0.86	0.98	0.54	0.69	0.39	0.54
$CV (t_{2})$	0.73	0.72	0.47	0.47	0.35	0.34
$Σ CV (t_{i})$	1.59	1.70	1.01	1.16	0.74	0.88

Table 4.6
Results of formulation D for the $M u n i c S w$ population
Table summary
This table displays the results of Results of formulation D for the $M u n i c S w$ population. The information is grouped by XXXX
XXXX
XXXX (appearing as row headers), 290
10%, 579
20% and 869
30%, calculated using BSSM-D and Textbook units of measure (appearing as column headers).
$n$ $S a m p l i n g f r a c t i o n$ $R e s u l t$	290 10%		579 20%		869 30%
$n$ $S a m p l i n g f r a c t i o n$ $R e s u l t$	BSSM-D	Textbook	BSSM-D	Textbook	BSSM-D	Textbook
$n_{1}$	67	59	134	118	202	182
$n_{2}$	68	77	136	153	206	233
$n_{3}$	40	35	80	70	120	107
$n_{4}$	58	47	116	93	171	128
$n_{5}$	32	43	65	85	97	129
$n_{6}$	16	21	31	43	47	65
$n_{7}$	9	8	17	17	26	25
$CV (t_{1})$	5.93	5.40	4.01	3.61	3.10	2.75
$CV (t_{2})$	12.53	12.24	8.36	8.12	6.36	6.14
$CV (t_{3})$	19.49	20.19	12.46	13.01	8.95	9.56
$CV (t_{4})$	16.91	17.45	10.85	11.27	7.84	8.30
$Σ CV (t_{i})$	54.86	55.28	35.68	36.01	26.25	26.75

As expected, in all three cases the sum of the coefficients of variation obtained by solving Formulation D were smaller than (bold) those obtained using the textbook algorithm. However, the textbook algorithm provided smaller CVs for some of the survey variables, in particular for the $M u n i c S w$ population. The improvements were generally not very large, but again were slightly larger for the $S c h o o l s N o r t h e a s t$ population. In this comparison, however, the allocations are quite different between the two methods.

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Integer programming formulations applied to optimal allocation in stratified sampling 4. Numerical resultsInteger programming formulations applied to optimal allocation in stratified sampling 4. Numerical results