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Capstone — a project portfolio

Two constraints, one binary objective.

Chapter: XV — Capstone · lesson id capstone-portfolio · practice lesson

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You have five candidate projects. Each has a value v_p, a cost c_p, and a team-size demand t_p. You have a budget of 15 and a team capacity of 7. Pick projects to maximise total value.

project 1: v=10, c=5, t=2 project 2: v= 7, c=3, t=1 project 3: v= 8, c=4, t=1 project 4: v= 5, c=2, t=1 project 5: v=12, c=6, t=3

Write the model yourself. The check below confirms your optimum equals 32.

Predict before running

Five projects, budget and team constraint. About how many will the solver pick?

• All five. Every project has positive value
• Around four. Drop the worst-fit project
• Just one or two. Budget is tight

Starter code

model portfolio
set P = {1, 2, 3, 4, 5}
const v = {10, 7, 8, 5, 12}, forall p in P
const c = {5, 3, 4, 2, 6}, forall p in P
const t = {2, 1, 1, 1, 3}, forall p in P
# TODO: declare bin x indexed over P, the budget constraint (sum c*x <= 15),
#       the team constraint (sum t*x <= 7), and the max sum v*x objective.
end

import elytica
def main():
  elytica.init_model("portfolio")
  elytica.run_model("portfolio")
  for i in [1,2,3,4,5]:
    print("x"+str(i), "=", elytica.get_variable_value("portfolio", "x"+str(i)))
  print("obj =", elytica.get_best_primal_bound("portfolio"))
  return 0

Hint

Five binary x_p variables. Two ≤ constraints (budget and team). One max sum objective.

What counts as correct

• no-error — Model solves
• numeric-match — Optimal portfolio value is 32 (projects 2, 3, 4, 5)
• code-contains — Reads the objective from the solver (no hardcoding)

Solution

model portfolio
set P = {1, 2, 3, 4, 5}
const v = {10, 7, 8, 5, 12}, forall p in P
const c = {5, 3, 4, 2, 6}, forall p in P
const t = {2, 1, 1, 1, 3}, forall p in P
bin x, forall p in P
constr sum_{p in P}{c_{p} * x_{p}} <= 15
constr sum_{p in P}{t_{p} * x_{p}} <= 7
max sum_{p in P}{v_{p} * x_{p}}
end

import elytica
def main():
  elytica.init_model("portfolio")
  elytica.run_model("portfolio")
  for i in [1,2,3,4,5]:
    print("x"+str(i), "=", elytica.get_variable_value("portfolio", "x"+str(i)))
  print("obj =", elytica.get_best_primal_bound("portfolio"))
  return 0

Mastery variant

Five new projects with different values, costs, and teams. Pick the best portfolio.

model portfolio_v
set P = {1, 2, 3, 4, 5}
const v = {8, 6, 9, 4, 11}, forall p in P
const c = {4, 3, 5, 2, 5}, forall p in P
const t = {2, 2, 2, 1, 3}, forall p in P
# TODO: bin x indexed over P, budget (sum c*x <= 13), team (sum t*x <= 8),
#       and max sum v*x.
end

import elytica
def main():
  elytica.init_model("portfolio_v")
  elytica.run_model("portfolio_v")
  for i in [1,2,3,4,5]:
    print("x"+str(i), "=", elytica.get_variable_value("portfolio_v", "x"+str(i)))
  print("obj =", elytica.get_best_primal_bound("portfolio_v"))
  return 0

Read this through MCP: read with path: "lessons/capstone-portfolio", or resource hlpl://lessons/capstone-portfolio.