How to Use Mathematical Modeling to Optimize Everyday Decisions

1. Time Management – 0‑1 Knapsack Model

Define N daily tasks. Task i has importance v_i, required time t_i, and satisfaction gain s_i. With total available time T, the optimization problem is

max   Σ s_i·x_i
s.t.  Σ t_i·x_i ≤ T
      x_i ∈ {0,1}   (i = 1,…,N)

where x_i indicates whether task i is selected. The problem is a classic 0‑1 knapsack and can be solved by dynamic programming. A concise Python implementation is:

def knapsack(values, times, T):
    dp = [0]*(T+1)
    for v,t in zip(values, times):
        for w in range(T, t-1, -1):
            dp[w] = max(dp[w], dp[w-t] + v)
    return dp[T]

To reflect human energy cycles, introduce a time‑weight factor w(t). Effective satisfaction becomes s_i·w(t_i). Empirical studies suggest weights of 1.2–1.5 for 10 am–12 pm and 3 pm–5 pm, and 0.8–1.0 for other periods.

2. Shopping Decision – Cost‑Benefit Index and Discounted Cash‑Flow

The cost‑benefit index for a product is defined as <code>I = (U × L) / P</code> where U is a usage‑value score (1–10), L the expected lifespan in years, and P the purchase price. Example: a 200 CNY shirt (U=8, L=2) yields I=0.08 ; a 500 CNY coat (U=9, L=5) yields I=0.09 , indicating higher value despite higher cost. For bulk or long‑term purchases, apply a discounted cash‑flow model. Let P be the upfront price, S the annual cost saving, N the useful life (years), and r the discount rate. The net present value is <code>NPV = Σ_{k=1}^{N} S / (1+r)^k – P</code> The investment is attractive when NPV > 0 . 3. Commuting Route – Graph Model and Shortest‑Path Algorithms Model the city’s road network as a directed weighted graph G = (V, E) . For each edge e ∈ E define a composite weight <code>w_e = α·time_e + β·cost_e + γ·inconvenience_e + δ·preference_e</code> where the coefficients α,β,γ,δ reflect the decision‑maker’s priorities. The optimal route from home to work is the path with minimal total weight, obtainable with Dijkstra’s algorithm (for static weights) or A* (when heuristic estimates of remaining travel time are available). Collect a week of travel‑time data, calibrate the coefficients, and optionally make w_e time‑dependent to capture rush‑hour effects. 4. Health Management – Nutrition Linear Programming and Exercise Energy Model Nutrition planning is a linear programming (LP) problem. Let M be the number of foods and N the number of nutrients. Define: a_{ij} : amount of nutrient j in one unit of food i c_i : cost of one unit of food i d_j : daily requirement of nutrient j x_i : quantity of food i to purchase The LP formulation is <code>min Σ_{i=1}^{M} c_i·x_i s.t. Σ_{i=1}^{M} a_{ij}·x_i ≥ d_j for j = 1,…,N x_i ≥ 0</code> Solving yields the cheapest diet that satisfies all nutritional constraints. For instance, meeting 2000 kcal, 60 g protein, and 300 g carbohydrate per day can be expressed directly in the model. Exercise planning can use a heart‑rate‑intensity‑calorie relationship. Target heart‑rate zone: <code>HR_target = 0.6–0.85 × (220 – age)</code> Weekly calorie‑burn goal C_goal (e.g., 1500–2500 kcal). Daily burn estimate: <code>C_day = Σ_{sessions} k·(HR_avg – HR_rest)·duration</code> where k converts heart‑rate surplus and duration into kilocalories. Adjust dietary intake so that total intake – total burn ≈ desired weight change. 5. Financial Planning – Markowitz Mean‑Variance Portfolio Optimization Assume n assets with expected returns μ_i and covariance matrix Σ . Decision variables are portfolio weights w_i (Σ w_i = 1, w_i ≥ 0). Portfolio expected return and variance are <code>R_p = Σ w_i·μ_i σ_p^2 = w^T·Σ·w</code> Two common formulations: Maximize R_p subject to σ_p^2 ≤ σ_target . Minimize σ_p^2 subject to R_p ≥ R_target . These quadratic programs are solved with standard QP solvers, producing the efficient frontier. The key insight is diversification: spreading capital reduces variance without sacrificing expected return. 6. Multi‑Objective Decision Analysis – AHP and Weighted Scoring When evaluating m alternatives against k criteria, assign a score s_{ij} to alternative i on criterion j . Derive criterion weights w_j (Σ w_j = 1) using the Analytic Hierarchy Process (AHP): construct a pairwise‑comparison matrix, compute its principal eigenvector, and normalize. The overall score for alternative i is <code>S_i = Σ_{j=1}^{k} w_j·s_{ij}</code> Select the alternative with the highest S_i . This method quantifies trade‑offs such as housing price, commute time, education quality, and environmental factors, providing a transparent basis for complex life decisions.