# Multiobjective sample allocation for constraint multivariate and multidomain optimal allocation in survey sampling (a stepwise optimality procedure is processed first to force Pareto optimality of the solution) Computes solutions to standard sample allocation problems under various precision and cost restrictions. The input data is transformed and parsed to the Embedded COnic Solver (ECOS) from the 'ECOSolveR' package. Multiple survey purposes are optimized simultaneously through a stepwise weighted Chebyshev minimization which forces Pareto optimality of solutions (cf. `mosalloc()`). ```r mosallocStepwiseFirst( D, d, A = NULL, a = NULL, C = NULL, c = NULL, l = 2, u = NULL, opts = list(sense = "max_precision", init_w = 1, mc_cores = 1L, max_iters = 100L) ) ``` ## Arguments - `D`: (type: matrix) The objective matrix. A matrix of either precision or cost units. - `d`: (type: vector) The objective vector. A vector of either fixed precision components (e.g. finite population corrections) or fixed costs. - `A`: (type: matrix) A matrix of precision units for precision constraints. - `a`: (type: vector) The right-hand side vector of the precision constraints. - `C`: (type: matrix) A matrix of cost coefficients for cost constraints - `c`: (type: vector) The right-hand side vector of the cost constraints. - `l`: (type: vector) A vector of lower box constraints. - `u`: (type: vector) A vector of upper box constraints. - `opts`: (type: list) The options used by the algorithms: `$sense` (type: character) Sense of optimization (default = `"max_precision"`, alternative `"min_cost"`). `$init_w` (type numeric or matrix) Preference weightings (default = 1; The weight for first objective component must be 1). `$mc_cores` (type: integer) The number of cores for parallelizing multiple input weightings stacked rowwise (default = 1L). `max_iters` (type: integer) The maximum number of iterations (default = 100L). ## Returns The function `mosallocStepwiseFirst()` returns a list containing the following components: `$w` The initial preference weighting `opts$init_w`. `$n` The vector of optimal sample sizes. `$J` The optimal objective vector. `$Objective` The objective value with respect to decision functional f. `NULL` if `opts$f = NULL`. `$Utopian` Always `NULL` (consistency to `mosalloc()` output). `NULL` if `opts$f = NULL`. `$Normal` The vector normal to the Pareto frontier at `$J`. `$dfJ` Always `NULL` (consistency to `mosalloc()` output). `$Sensitivity` The dual variables of the objectives and constraints. `$Qbounds` The Quality bounds of the Lorentz cones. `$Dbounds` The weighted objective constraints ($w * $J). `$Scalepar` An internal scaling parameter. `$Ecosolver` A list of ECOSolveR returns including: `...$Ecoinfostring` The info string of `ECOSolveR::ECOS_csolve()`. `...$Ecoredcodes` The redcodes of `ECOSolveR::ECOS_csolve()`. `...$Ecosummary` Problem summary of `ECOSolveR::ECOS_csolve()`. `$Timing` Run time info. `$Iteration` Always `NULL` (consistency to `mosalloc()` output). ## Examples ```r # Artificial population of 50 568 business establishments and 5 business # sectors (data from Valliant, R., Dever, J. A., & Kreuter, F. (2013). # Practical tools for designing and weighting survey samples. Springer. # https://doi.org/10.1007/978-1-4614-6449-5, Example 5.2 pages 133-9) # See also https://umd.app.box.com/s/9yvvibu4nz4q6rlw98ac/file/297813512360 # file: Code 5.3 constrOptim.example.R Nh <- c(6221, 11738, 4333, 22809, 5467) # stratum sizes ch <- c(120, 80, 80, 90, 150) # stratum-specific cost of surveying # Revenues mh.rev <- c(85, 11, 23, 17, 126) # mean revenue Sh.rev <- c(170.0, 8.8, 23.0, 25.5, 315.0) # standard deviation revenue # Employees mh.emp <- c(511, 21, 70, 32, 157) # mean number of employees Sh.emp <- c(255.50, 5.25, 35.00, 32.00, 471.00) # std. dev. employees # Proportion of estabs claiming research credit ph.rsch <- c(0.8, 0.2, 0.5, 0.3, 0.9) # Proportion of estabs with offshore affiliates ph.offsh <- c(0.06, 0.03, 0.03, 0.21, 0.77) budget <- 300000 # overall available budget n.min <- 100 # minimum stratum-specific sample size #---------------------------------------------------------------------------- # Problem: Minimization of the maximum relative variation of estimates for # the total revenue, the number of employee, the number of businesses claimed # research credit and the number of businesses with offshore affiliates # subject to cost restrictions l <- rep(n.min, 5) # minimum sample size ber stratum u <- Nh # maximum sample size per stratum C <- rbind(ch, ch * c(-1, -1, -1, 0, 0)) c <- c(budget, - 0.5 * budget) A <- NULL # no precision constraint a <- NULL # no precision constraint # Variance components for multidimensional objective D <- rbind(Sh.rev**2 * Nh**2/sum(Nh * mh.rev)**2, Sh.emp**2 * Nh**2/sum(Nh * mh.emp)**2, ph.rsch * (1 - ph.rsch) * Nh**3/(Nh - 1)/sum(Nh * ph.rsch)**2, ph.offsh * (1 - ph.offsh) * Nh**3/(Nh - 1)/sum(Nh * ph.offsh)**2) d <- as.vector(D %*% (1 / Nh)) # finite population correction opts = list(sense = "max_precision", init_w = 1, mc_cores = 1L, max_iters = 100L) res1 <- mosallocStepwiseFirst(D = D, d = d, C = C, c = c, l = l, u = u, opts = opts) w <- res1$J[1] / res1$J w # [1] 1.000000 3.879692 2.653655 1.000000 opts = list(sense = "max_precision", init_w = w, mc_cores = 1L, max_iters = 100L) res2 <- mosallocStepwiseFirst(D = D, d = d, C = C, c = c, l = l, u = u, opts = opts) res2$w # [1] 1.000000 3.879692 2.653655 1.000000 # Compare to function mosalloc (without stepwise procedure) opts = list(sense = "max_precision", f = NULL, df = NULL, Hf = NULL, init_w = w, mc_cores = 1L, pm_tol = 1e-05, max_iters = 100L, print_pm = FALSE) res3 <- mosalloc(D = D, d = d, C = C, c = c, l = l, u = u, opts = opts) # Compare objectives rbind(res1$J, res2$J, res3$J) # [,1] [,2] [,3] [,4] #[1,] 0.00170589 0.0004396972 0.0006428453 0.00170589 #[2,] 0.00170589 0.0004396971 0.0006428420 0.00170589 #[3,] 0.00170589 0.0004396971 0.0006428440 0.00170589 # Compare optimal sample sizes rbind(res1$n, res2$n, res3$n) # [,1] [,2] [,3] [,4] [,5] # [1,] 958.0510 290.7446 147.1789 602.8856 638.2686 # [2,] 958.0455 290.7447 147.1871 602.8847 638.2692 # [3,] 958.0488 290.7446 147.1822 602.8853 638.2688 ``` ## References See: Willems, F. (2025). A Framework for Multiobjective and Uncertain Resource Allocation Problems in Survey Sampling based on Conic Optimization (Doctoral dissertation). Trier University. tools:::Rd_expr_doi("10.25353/ubtr-9200-484c-5c89") .

mosallocStepwiseFirst function