Add a weighted goal achievement approach for multi-objective optimization to a project problem (Jones and Tamiz 2010).
Arguments
- x
multi_problem()object.- weights
numericvector containing the weights for each objective. To generate multiple solutions based on different values,weightscan be anumericmatrix where each row corresponds to a different solution and each columns corresponds to a different objective.- goals
numericvector containing values that denote the reference points. These points represent aspirational goals for each objective. To generate multiple solutions based on different values,goalscan be anumericmatrix where each row corresponds to a different solution and each columns corresponds to a different objective. Note that all values must be greater than zero.- verbose
logicalshould progress on generating solutions displayed? Defaults toTRUE.
Value
A multi_problem() object with the approach added to it.
Details
The weighted goal achievement approach for multi-objective optimization involves creating a new objective that is calculated based on multiple objectives. In particular, the new objective uses weights to specify the relative importance of each individual objective, and goals to specify a threshold minimum level of performance for each objective (conceptually similar to target thresholds used in conservation planning). It then calculates the new objectives based on the weighted sum of the percentage of each goal that is achieved for each objective.
To describe this approach mathematically, we will define the following terminology. Let \(O\) denote the set of objectives (indexed by \(o\)). For each objective, let \(W_o\) denote the weight for each objective \(o \in O\), \(G_o\) denote the goal for each objective \(o \in O\), and \(V_o\) denote the objective value for a candidate solution as measured based on each objective \(o \in O\). After defining these terms, the approach is formulated with the following equation.
$$ \mathrm{Minimize} \space \sum_{o = 0}^{O} W_o \times \frac{V_o}{G_o} $$
References
Jones D and Tamiz M (2010) Goal Programming Variants. and Management Science, volume 141. Springer, Boston, MA.
See also
Other approaches:
add_abs_constraint_approach(),
add_ref_point_approach()
Examples
# load data
data(sim_multi_projects)
data(sim_multi_features)
data(sim_multi_actions)
data(sim_multi_tree)
# build problem
p <-
multi_problem(
obj1 =
problem(
sim_multi_projects[[1]], sim_multi_actions, sim_multi_features[[1]],
"name", "success", "name", "cost", "name",
baseline_project_name = "baseline_project_obj1"
) %>%
add_max_phylo_div_objective(
budget = 1000, tree = sim_multi_tree[[1]]
) %>%
add_binary_decisions(),
obj2 =
problem(
sim_multi_projects[[2]], sim_multi_actions, sim_multi_features[[2]],
"name", "success", "name", "cost", "name",
baseline_project_name = "baseline_project_obj2"
) %>%
add_max_richness_objective(budget = 1000) %>%
add_binary_decisions(),
obj3 =
problem(
sim_multi_projects[[3]], sim_multi_actions, sim_multi_features[[3]],
"name", "success", "name", "cost", "name",
baseline_project_name = "baseline_project_obj3"
) %>%
add_max_wtd_sum_objective(budget = 1000) %>%
add_binary_decisions()
) %>%
add_wtd_goal_approach(weights = c(10, 11, 12), goals = c(3, 4, 5)) %>%
add_default_solver()
# print problem
print(p)
#> Multi-objective Project Prioritization Problem
#> objective: obj1
#> projects: F1_project, F2_project, F8_project, baseline_project_obj1 (4 projects)
#> features: F1, F2, F8 (3 features)
#> project success: proportion values (between 0.832 and 1)
#> objective: maximum phylogenetic diversity objective
#> targets: none specified
#> weights: none specified
#> constraints: none specified
#> decisions: binary decision
#> objective: obj2
#> projects: F3_project, F4_project, baseline_project_obj2 (3 projects)
#> features: F3, F4 (2 features)
#> project success: proportion values (between 0.85 and 1)
#> objective: maximum richness objective
#> targets: none specified
#> weights: none specified
#> constraints: none specified
#> decisions: binary decision
#> objective: obj3
#> projects: F5_project, F6_project, F7_project, ... (6 projects)
#> features: F5, F6, F7, ... (5 features)
#> project success: proportion values (between 0.715 and 1)
#> objective: maximum weighted sum objective
#> targets: none specified
#> weights: none specified
#> constraints: none specified
#> decisions: binary decision
#> actions: A1_action, A2_action, A3_action, ... (18 actions)
#> action costs: continuous values (between 0 and 103.226)
#> approach: weighted goal approach
#> solver: gurobi solver
# solve problem
s <- solve(p)
#> Set parameter Username
#> Set parameter LicenseID to value 2806834
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter ScaleFlag to value 2
#> Set parameter NumericFocus to value 1
#> Set parameter Presolve to value 2
#> Set parameter Threads to value 1
#> Set parameter PoolSolutions to value 1
#> Set parameter PoolSearchMode to value 2
#> Academic license - for non-commercial use only - expires 2027-04-14
#> Gurobi Optimizer version 13.0.1 build v13.0.1rc0 (linux64 - "Ubuntu 24.04.2 LTS")
#>
#> CPU model: 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz, instruction set [SSE2|AVX|AVX2|AVX512]
#> Thread count: 4 physical cores, 8 logical processors, using up to 1 threads
#>
#> Non-default parameters:
#> TimeLimit 2147483647
#> MIPGap 0
#> ScaleFlag 2
#> NumericFocus 1
#> Presolve 2
#> Threads 1
#> PoolSolutions 1
#> PoolSearchMode 2
#>
#> Optimize a model with 335 rows, 263 columns and 1122 nonzeros (Min)
#> Model fingerprint: 0x9528ff18
#> Model has 3 linear objective coefficients
#> Variable types: 14 continuous, 150 integer (150 binary)
#> Semi-Variable types: 99 continuous, 0 integer
#> Coefficient statistics:
#> Matrix range [2e-02, 1e+02]
#> Objective range [1e+01, 1e+01]
#> Bounds range [6e-01, 3e+00]
#> RHS range [1e+00, 1e+03]
#>
#> Presolve removed 242 rows and 80 columns
#> Presolve time: 0.01s
#> Presolved: 289 rows, 281 columns, 928 nonzeros
#> Variable types: 101 continuous, 180 integer (180 binary)
#> Found heuristic solution: objective 25.5700391
#> Found heuristic solution: objective 24.6994718
#> Root relaxation presolved: 289 rows, 281 columns, 928 nonzeros
#>
#>
#> Root relaxation: objective 2.329846e+01, 132 iterations, 0.00 seconds (0.00 work units)
#>
#> Nodes | Current Node | Objective Bounds | Work
#> Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
#>
#> 0 0 23.29846 0 19 24.69947 23.29846 5.67% - 0s
#> 0 0 23.36824 0 19 24.69947 23.36824 5.39% - 0s
#> 0 0 23.92674 0 16 24.69947 23.92674 3.13% - 0s
#> 0 0 23.98388 0 19 24.69947 23.98388 2.90% - 0s
#> 0 0 24.09664 0 20 24.69947 24.09664 2.44% - 0s
#> 0 0 24.34277 0 27 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 31 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 31 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 29 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 16 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 22 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 22 24.69947 24.34277 1.44% - 0s
#> 0 0 24.34277 0 25 24.69947 24.34277 1.44% - 0s
#> 0 0 24.36402 0 21 24.69947 24.36402 1.36% - 0s
#> 0 0 24.40847 0 23 24.69947 24.40847 1.18% - 0s
#> 0 0 24.42298 0 24 24.69947 24.42298 1.12% - 0s
#> 0 0 24.42298 0 24 24.69947 24.42298 1.12% - 0s
#> 0 0 24.53181 0 19 24.69947 24.53181 0.68% - 0s
#> 0 0 24.54063 0 25 24.69947 24.54063 0.64% - 0s
#> 0 0 24.54063 0 27 24.69947 24.54063 0.64% - 0s
#> 0 0 24.54080 0 25 24.69947 24.54080 0.64% - 0s
#> 0 0 24.54086 0 23 24.69947 24.54086 0.64% - 0s
#> 0 0 24.54093 0 25 24.69947 24.54093 0.64% - 0s
#> 0 0 24.54093 0 26 24.69947 24.54093 0.64% - 0s
#> 0 0 24.54101 0 24 24.69947 24.54101 0.64% - 0s
#> 0 0 24.54101 0 26 24.69947 24.54101 0.64% - 0s
#> 0 0 24.54102 0 26 24.69947 24.54102 0.64% - 0s
#> 0 0 24.54102 0 26 24.69947 24.54102 0.64% - 0s
#> 0 2 24.58380 0 26 24.69947 24.58380 0.47% - 0s
#>
#> Cutting planes:
#> Cover: 9
#> MIR: 2
#> RLT: 2
#> Relax-and-lift: 1
#>
#> Explored 29 nodes (696 simplex iterations) in 0.08 seconds (0.06 work units)
#> Thread count was 1 (of 8 available processors)
#>
#> Solution count 1: 24.6995
#> No other solutions better than 24.6995
#>
#> Optimal solution found (tolerance 0.00e+00)
#> Best objective 2.469947178589e+01, best bound 2.469947178589e+01, gap 0.0000%
# print solution
print(s)
#> # A tibble: 1 × 47
#> solution status cost obj1 obj2 obj3 A1_action A2_action A3_action
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <lgl> <lgl> <lgl>
#> 1 1 OPTIMAL 971. 0.420 1.02 1.75 TRUE FALSE TRUE
#> # ℹ 38 more variables: A4_action <lgl>, A5_action <lgl>, A6_action <lgl>,
#> # A7_action <lgl>, A8_action <lgl>, A9_action <lgl>, A10_action <lgl>,
#> # A11_action <lgl>, A12_action <lgl>, A13_action <lgl>, A14_action <lgl>,
#> # A15_action <lgl>, B1_action <lgl>, B2_action <lgl>, B3_action <lgl>,
#> # F1_project <lgl>, F2_project <lgl>, F8_project <lgl>,
#> # baseline_project_obj1 <lgl>, F3_project <lgl>, F4_project <lgl>,
#> # baseline_project_obj2 <lgl>, F5_project <lgl>, F6_project <lgl>, …