Designing the promotion budget mix across digital and outdoor media Based on Simulated Annealing

Project info

• Field of studyMarketing and Consumer Behaviour
• LevelDoctoral dissertation (S3)
• Method typeOptimisation
• Estimated build time± 7 working days
• Toolsnumpy (implementasi manual); scipy.optimize.dual_annealing sebagai pembanding
• PriceUS$34 - US$206

Research goal

Apply Simulated Annealing to synthetic data to find the best solution for the promotion budget mix across digital and outdoor media and compare outcomes.

Variables

Example result

• 16.348Best objective
• 16.339Global optimum
• 0.056Gap to optimum %
• 0.586Acceptance ratio
• 500Iterations
• 0.081Final temperature

Simulated annealing (500 iterations) lowers Consumer reach to 16.35 at Digital media budget = 119.76 and Outdoor media budget = 68.95; 58.6% of moves were accepted and the gap to the optimum is 0.06%.

Computed from illustrative synthetic data, not from a real study.

About Simulated Annealing

Simulated annealing explores the solution space by accepting worse moves with a probability that falls as temperature decreases. It suits global optimisation of functions with many local minima and combinatorial problems.

Steps

1. 1Define the objective, bounds, and starting point
2. 2Set the initial temperature and cooling schedule
3. 3Generate a random neighbour solution
4. 4Accept if better, or with probability exp(-delta/T) if worse
5. 5Lower the temperature and repeat
6. 6Plot the objective trajectory and report the best solution

Assumptions to check

• The cooling schedule is slow enough to avoid local minima
• The neighbour step matches the problem scale
• Several runs with different seeds assess consistency

Reading and reporting the results

Report the initial temperature, cooling rate, best solution and objective value, and their variation across runs. Explain that results are heuristic with no guarantee of the global optimum.

Example Python code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import dual_annealing

def f(p):
    return (p[0] ** 2 + p[1] ** 2) / 10 + 3 * (2 - np.cos(2 * p[0]) - np.cos(2 * p[1]))

rng = np.random.default_rng(32)
digital_media_budget = rng.uniform(-5, 5, 2)          # current solution
cur = f(digital_media_budget)
best_x, best = digital_media_budget.copy(), cur
T, cooling = 5.0, 0.995
consumer_reach = []                              # objective trajectory
for it in range(4000):
    cand = np.clip(digital_media_budget + rng.normal(0, 0.5, 2), -5, 5)
    fc = f(cand)
    if fc < cur or rng.random() < np.exp(-(fc - cur) / T):
        digital_media_budget, cur = cand, fc
        if cur < best:
            best_x, best = digital_media_budget.copy(), cur
    consumer_reach.append(best)
    T *= cooling

print("SA best x:", best_x.round(4), "f =", round(best, 5))
ref = dual_annealing(f, bounds=[(-5, 5), (-5, 5)], seed=32)
print("scipy dual_annealing:", ref.x.round(4), "f =", round(ref.fun, 5))

plt.plot(consumer_reach)
plt.xlabel("Iteration"); plt.ylabel("Best objective")
plt.show()

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions