Prescriptive Analytics

Prescriptive analytics represents the most advanced form of analytics, focusing on "what should we do" by providing actionable recommendations through optimization algorithms, simulation, and decision theory. In automotive applications, prescriptive analytics drives strategic decisions in pricing, inventory management, marketing allocation, and operational optimization.

Mathematical Foundation

Prescriptive analytics combines optimization theory, decision science, and simulation to recommend optimal actions:

Linear Programming

Standard Form

Linear programming solves optimization problems with linear objective functions and constraints:

Where:

• is the objective function coefficient vector
• is the decision variable vector
• is the constraint coefficient matrix
• is the constraint bounds vector

Simplex Method

The simplex algorithm iteratively moves between vertices of the feasible region:

Where is the search direction and is the step size.

Automotive Example: Production Planning Optimization

Business Context: An automotive manufacturer needs to optimize production across multiple plant locations to minimize cost while meeting demand.

Decision Variables:

• = Number of vehicles of type produced at plant

Objective Function (Minimize Total Cost):

Constraints:

Demand Satisfaction:

Plant Capacity:

Non-negativity:

Sample Problem:

• Vehicles: Sedan (S), SUV (U), Truck (T)
• Plants: Detroit (D), Atlanta (A), Los Angeles (L)
• Production costs: , ,

Mathematical Formulation:

Solution: Optimal production allocation minimizes total cost while satisfying all constraints.

Business Impact: 12% cost reduction through optimal plant utilization.

Integer Programming

Mathematical Formulation

Integer programming extends linear programming with integer constraints:

Binary Integer Programming

For yes/no decisions:

Automotive Example: Dealership Location Selection

Business Context: An automotive group wants to select optimal locations for new dealerships to maximize market coverage while staying within budget.

Decision Variables:

• if location is selected, 0 otherwise

Objective Function (Maximize Market Coverage):

Where is the market potential at location .

Budget Constraint:

Market Coverage Constraint:

Where represents locations that can serve market segment .

Binary Constraint:

Business Results: Optimal selection of 8 locations from 25 candidates, achieving 95% market coverage within budget.

Network Optimization

Minimum Cost Flow Problem

Network flow problems optimize flow through networks:

Where:

• is the cost per unit flow on arc
• is the flow on arc
• is the supply/demand at node
• is the capacity of arc

Automotive Example: Supply Chain Optimization

Business Context: An automotive parts supplier optimizes distribution from factories to dealers to minimize transportation costs.

Network Structure:

• Supply Nodes: 3 factories with production capacity
• Demand Nodes: 50 dealerships with parts demand
• Transshipment Nodes: 8 distribution centers

Mathematical Model:

Flow Conservation at each node:

Capacity Constraints:

Cost Minimization:

Where is the distance between nodes and .

Business Impact: 18% reduction in transportation costs through optimal routing.

Dynamic Programming

Bellman Equation

Dynamic programming solves multi-stage decision problems:

Where:

• is the value function at state and time
• is the immediate reward
• is the discount factor
• is the next state given action

Automotive Example: Inventory Management Optimization

Business Context: A dealership optimizes parts inventory ordering to minimize holding and stockout costs.

State Variables:

• = Current inventory level at time

Decision Variables:

• = Order quantity at time

Cost Function:

Where:

• = ordering cost per unit
• = holding cost per unit
• = penalty cost per unit shortage
• = demand at time

Bellman Equation:

Optimal Policy: policy where:

• If inventory , order up to
• If inventory , don't order

Business Results: 25% reduction in total inventory costs while maintaining 98% service level.

Game Theory

Nash Equilibrium

In competitive environments, game theory provides optimal strategies:

Pure Strategy Nash Equilibrium: Strategy profile where:

Mixed Strategy Equilibrium

When no pure strategy equilibrium exists:

Where is the probability distribution over strategies.

Automotive Example: Competitive Pricing Strategy

Business Context: Two automotive dealerships compete on pricing for the same vehicle model.

Payoff Matrix (Daily Profit in 000s):

Nash Equilibrium Analysis:

Pure Strategy: (High, High) with payoffs (10, 10) Mixed Strategy: Each player plays High with probability

Indifference Condition:

Solving:

Expected Payoff:

Business Strategy: Implement high pricing 60% of the time for optimal expected profit.

Simulation and Monte Carlo Methods

Monte Carlo Simulation

For complex systems with uncertainty:

Where are random samples from the distribution of .

Confidence Interval:

Automotive Example: Service Center Capacity Planning

Business Context: A service center uses simulation to determine optimal staffing levels considering uncertain arrival patterns and service times.

Model Parameters:

• Arrival Process: Poisson with rate customers/hour
• Service Time: Exponential with mean minutes
• Service Bays: (decision variable)

Performance Metrics:

Utilization:

Average Wait Time (M/M/s queue):

Monte Carlo Implementation:

1. Generate simulation runs
2. For each run, simulate daily operations
3. Calculate performance metrics
4. Estimate expected values and confidence intervals

Results:

• 4 Service Bays: Average wait = 8.5 minutes, Utilization = 75%
• 5 Service Bays: Average wait = 3.2 minutes, Utilization = 60%
• 6 Service Bays: Average wait = 1.8 minutes, Utilization = 50%

Optimal Decision: 5 service bays balance cost and customer satisfaction.

Decision Analysis

Expected Value Calculation

For decisions under uncertainty:

Where is the probability and is the value of outcome .

Expected Value of Perfect Information (EVPI)

Automotive Example: New Model Launch Decision

Business Context: An automotive manufacturer decides whether to launch a new electric vehicle model considering market uncertainty.

Decision Alternatives:

• Launch: High investment, uncertain returns
• Don't Launch: No investment, no returns
• Market Research: Additional information at cost

Market Scenarios:

• Strong Market (P = 0.4): High EV adoption
• Moderate Market (P = 0.5): Gradual EV adoption
• Weak Market (P = 0.1): Slow EV adoption

Payoff Matrix (NPV in millions):

Expected Value Calculation:

Decision: Launch the new model (EV = 0M).

Value of Perfect Information:

Since EVPI < 0, perfect information has no additional value.

Multi-Criteria Decision Analysis (MCDA)

Weighted Sum Model

Where:

• is the overall score for alternative
• is the weight for criterion
• is the normalized score of alternative on criterion

Analytic Hierarchy Process (AHP)

Pairwise Comparison Matrix:

Priority Vector (eigenvalue method):

Consistency Ratio:

Automotive Example: Supplier Selection

Business Context: An automotive manufacturer selects a battery supplier for electric vehicles using multiple criteria.

Selection Criteria:

• Cost (Weight: 0.30)
• Quality (Weight: 0.35)
• Delivery (Weight: 0.20)
• Innovation (Weight: 0.15)

Supplier Alternatives:

• Supplier A: Cost-focused option
• Supplier B: Quality-focused option
• Supplier C: Balanced option

Decision Matrix (normalized scores):

Weighted Score Calculation for Supplier B:

Decision: Suppliers B and C tie for best overall score. Additional analysis needed.

Stochastic Programming

Two-Stage Stochastic Programming

For decisions under uncertainty with recourse actions:

Where:

Automotive Example: Fleet Capacity Planning Under Demand Uncertainty

Business Context: A car rental company plans fleet size considering uncertain seasonal demand.

First-Stage Decision (): Fleet size by vehicle type Second-Stage Decision (): Additional rentals or idle capacity

Objective Function:

Where:

• = annual cost per vehicle type
• = penalty cost for unmet demand
• = holding cost for excess capacity
• = unmet demand, = excess capacity

Demand Scenarios: High season (P=0.3), Normal (P=0.5), Low season (P=0.2)

Optimal Solution: Fleet composition balances fixed costs with expected penalty/holding costs across all scenarios.

Automotive Industry Applications

Auto Finance

• Portfolio Optimization: Risk-return optimization for loan portfolios
• Credit Allocation: Optimal credit limits using stochastic programming
• Interest Rate Strategy: Dynamic programming for rate setting

Auto Marketing

• Budget Allocation: Multi-channel marketing optimization
• Customer Targeting: Integer programming for campaign selection
• Price Optimization: Game-theoretic competitive pricing

Auto Sales

• Inventory Management: Stochastic inventory models
• Territory Planning: Network optimization for sales territories
• Incentive Design: Mechanism design for sales compensation

Dealer Operations

• Service Scheduling: Optimization of service bay utilization
• Parts Inventory: Dynamic programming for parts ordering
• Facility Layout: Operations research for optimal layout design

Prescriptive analytics transforms data insights into actionable decisions, providing automotive organizations with mathematically optimal strategies for complex business challenges. By leveraging optimization theory, decision science, and simulation techniques, companies can achieve measurable improvements in efficiency, profitability, and customer satisfaction.