Chapter 5: Prescriptive Analytics & Optimization

# Chapter 5: Prescriptive Analytics & Optimization Prescriptive analytics takes the insights from predictive models one step further—**it tells you what to do**. By framing decisions as optimization problems, simulation experiments, or reinforcement‑learning policies, data scientists can recommend concrete actions that respect business constraints and maximize strategic objectives. ## 5.1 What is Prescriptive Analytics? | Term | Definition | Typical Use‑Case | |------|------------|-----------------| | Prescriptive Analytics | The science of *suggesting* optimal actions based on data and constraints. | Optimizing inventory levels, fleet routing, dynamic pricing, staffing schedules. | | Decision Model | A formal representation of the decision problem, including variables, constraints, and an objective. | Linear programming model for supply‑chain allocation. | | Optimization | Finding the best solution (or a good approximation) to a decision model. | Solving a portfolio allocation that maximizes Sharpe ratio. | | Simulation | Running a computational experiment to explore *what‑if* scenarios without solving a deterministic model. | Monte‑Carlo inventory simulation under demand uncertainty. | | Reinforcement Learning (RL) | An ML framework where an agent learns a policy through trial‑and‑error interactions with an environment. | Dynamic pricing where the agent learns price‑elasticity over time. | Prescriptive analytics sits at the intersection of **operations research**, **simulation science**, and **machine learning**. It requires clear business framing, robust data, and rigorous experimentation—exactly the principles we reinforced in Chapter 4. ## 5.2 Linear Programming (LP) and Integer Programming (IP) ### 5.2.1 Problem Formulation An LP problem has the general form: text maximize cᵀx subject to Ax ≤ b x ≥ 0 * **Decision variables** `x` represent quantities to decide (e.g., units to produce, seats to allocate). * **Objective vector** `c` captures the value or cost associated with each decision. * **Constraint matrix** `A` and **right‑hand side** `b` encode capacity, budget, or policy limits. #### Example: Airline Seat Allocation | Variable | Meaning | |----------|---------| | `x₁` | Economy seats sold in Flight A | | `x₂` | Business seats sold in Flight A | | `x₃` | Economy seats sold in Flight B | | `x₄` | Business seats sold in Flight B | **Objective:** Maximize revenue max 200x₁ + 400x₂ + 210x₃ + 420x₄ **Constraints:** Seat limits and minimum business seats 80x₁ + 100x₂ ≤ 200 (Flight A capacity) 90x₃ + 120x₄ ≤ 250 (Flight B capacity) x₂, x₄ ≥ 0.10(x₁+x₂) , 0.10(x₃+x₄) (Business ≥10% of total) This LP can be solved in seconds with solvers like **Gurobi**, **CPLEX**, or open‑source **PuLP**. ### 5.2.2 From LP to Mixed‑Integer Programming (MIP) When decisions are discrete (e.g., assigning staff to shifts), **integer variables** are introduced: xᵢ ∈ {0,1} MIP is NP‑hard, but modern solvers handle thousands of binary variables for medium‑sized problems. ### 5.2.3 Practical Tips | Tip | Why it matters | |-----|----------------| | Use **basis** to understand variable activity | Reveals which constraints are binding and why a solution changed after a data update. | | Perform **sensitivity analysis** | Shows how objective changes with a parameter shift—critical for risk‑aware decision making. | | Keep models **modular** | Separates business rules from data ingestion; easier to maintain as policies evolve. | ## 5.3 Simulation Techniques Simulation lets you test *dynamic* and *uncertain* environments where closed‑form optimization is infeasible. ### 5.3.1 Monte‑Carlo Simulation 1. Define a probability distribution for uncertain inputs (e.g., demand). 2. Sample many scenarios. 3. Run the deterministic decision model for each scenario. 4. Aggregate results (mean, variance, percentiles). **Tooling:** NumPy, Pandas, and `scipy.stats` for sampling. python import numpy as np # Sample demand from a lognormal distribution demand = np.random.lognormal(mean=4, sigma=0.5, size=10_000) # Compute profits for a fixed order quantity q q = 100 profit = np.maximum(0, (price * np.minimum(demand, q)) - (cost * q)) print('Expected profit:', profit.mean()) ### 5.3.2 Discrete‑Event Simulation Useful when the system has queues, service times, or stochastic events—e.g., call centers, manufacturing lines. **Frameworks:** SimPy, Arena, AnyLogic. python import simpy def customer(env, name, service_time): yield env.timeout(service_time) env = simpy.Environment() for i in range(10): env.process(customer(env, f'Customer {i}', np.random.exponential(5))) env.run(until=100) ### 5.3.3 When to Use Simulation | Scenario | Recommended Method | |----------|--------------------| | Static capacity planning | LP/IP | | Demand uncertainty with simple decision rule | Monte‑Carlo | | Complex queuing systems | Discrete‑Event | | Adaptive policies that learn from feedback | RL | ## 5.4 Reinforcement Learning for Dynamic Decision‑Making RL frames decision making as a **Markov Decision Process (MDP)**: - **State** `s_t` captures the current environment (e.g., inventory level, time of day). - **Action** `a_t` is the decision (e.g., price set, stock level). - **Reward** `r_t` is the immediate profit or cost. - **Policy** `π(a|s)` maps states to actions. The goal is to learn a policy that maximizes the expected cumulative reward. ### 5.4.1 Classic RL Algorithms | Algorithm | Type | Use‑Case | |-----------|------|----------| | Q‑Learning | Model‑free | Small discrete action spaces | | SARSA | Model‑free | On‑policy control | | DQN | Deep RL | High‑dimensional state space | | PPO | Policy gradient | Stable training for large problems | ### 5.4.2 Practical Example: Dynamic Pricing python import gym import stable_baselines3 as sb3 # Custom environment wrapper class PricingEnv(gym.Env): def __init__(self): super().__init__() self.price_range = [10, 20, 30, 40] self.action_space = gym.spaces.Discrete(len(self.price_range)) self.observation_space = gym.spaces.Box(low=0, high=1, shape=(1,)) # e.g., time of day def step(self, action): price = self.price_range[action] demand = np.random.poisson(lam=20 - price) # Simple inverse relationship reward = price * demand done = False return np.array([0]), reward, done, {} env = PricingEnv() model = sb3.PPO("MlpPolicy", env, verbose=1) model.learn(total_timesteps=10_000) ### 5.4.3 Deployment Considerations - **Reward shaping** must align with business KPIs. - **Exploration vs. exploitation** trade‑offs can be tuned with ε‑greedy or entropy bonuses. - **Model drift**: retrain periodically as demand patterns change. - **Explainability**: augment RL with feature importance or rule extraction to satisfy governance. ## 5.5 Integrating Prescriptive Models into the Decision Cycle | Phase | Prescriptive Activity | Tooling | Key Output | |-------|-----------------------|---------|------------| | Problem Definition | Map business constraints to variables | Stakeholder workshops, decision canvas | Decision matrix | | Data Preparation | Forecast demand, estimate costs | Pandas, Prophet, sklearn | Feature set | | Modeling | Build LP/IP, set up simulation, train RL | PuLP, SimPy, Stable Baselines | Optimal policies | | Deployment | API, dashboard, automated pipelines | Flask, FastAPI, Airflow | Decision engine | | Impact Measurement | Track KPIs vs. baseline | Tableau, Power BI | ROI table | ### 5.5.1 Experimentation Framework Prescriptive analytics benefits from an **experiment‑oriented** mindset—just like Chapter 4: 1. **Hypothesis Canvas** – Frame the *why* of each decision. 2. **Design Document** – Capture constraints, data sources, evaluation metrics. 3. **A/B or Multi‑armed Bandit Tests** – Compare policy variants in production. 4. **Monitoring Dashboard** – Continuous tracking of objective function values. ## 5.6 Real‑World Case: Optimizing Distribution in a Global Retailer | Stage | Approach | Result | |-------|----------|--------| | Problem | Reduce shipping costs while maintaining service levels. | Targeted cost savings. | | Data | Historical orders, lead times, shipping rates. | Cleaned and imputed demand. | | Model | Mixed‑integer program for warehouse‑to‑store routes. | 12% cost reduction. | | Validation | Monte‑Carlo simulation of demand spikes. | 95% SLA maintained. | | Deployment | REST API integrated into order‑routing system. | Real‑time routing decisions. | | Impact | $3M annual savings; 15% reduction in carbon footprint. | Tangible strategic value. | ## 5.7 Practical Checklist for Prescriptive Analytics Projects | Checklist Item | Why It Matters | Suggested Tool/Practice | |-----------------|----------------|------------------------| | Clear objective | Avoids “model chasing” | Define KPI, success metric upfront | | Constraint validation | Ensures feasibility | Constraint sanity checks, solver diagnostics | | Sensitivity analysis | Understand risk | `scipy.optimize.linprog` sensitivity, MIP solvers’ shadow prices | | Robust data pipeline | Prevents stale inputs | Airflow DAGs, dbt models | | Explainability layer | Builds trust | SHAP for LP, policy extraction for RL | | Governance review | Meets regulatory compliance | Data lineage, audit logs | ## 5.8 Takeaways 1. **Prescriptive analytics turns data into action** by formally encoding business goals as mathematical programs. 2. **Linear programming** is fast and interpretable; **simulation** handles stochastic, dynamic systems; **reinforcement learning** offers adaptive, data‑driven policies for complex environments. 3. **Experimentation remains key**—even in optimization, you must validate assumptions, monitor performance, and iterate. 4. **Alignment between business, data, and model**—captured early in the hypothesis canvas—ensures that the prescribed actions deliver real, measurable value. --- **Next Chapter**: *Natural Language & Unstructured Data for Competitive Intelligence*—where we’ll explore how text mining and sentiment analysis can feed into the decision models we just built.