Multi-Objective Optimization

Lecture 8 - Management Science

Introduction

Client Briefing: EcoExpress

Operations Director’s Dilemma:

“EU regulations demand 40% emission cuts, but we can’t sacrifice profitability, service quality, or reliability!”

The Fleet Challenge

EcoExpress operates regional last-mile delivery across 3 cities

• EU Green Deal: 40% emission reduction by 2025
• Rising fuel costs (€2.1/L diesel)
• Amazon entering our market (speed pressure)
• Driver shortage (need automation-friendly vehicles)

Question: How do we transform our fleet while staying competitive?

Today’s Learning Objectives

By the end of this lecture, you will be able to:

1. Explain why most decisions involve competing objectives
2. Identify and visualize Pareto optimal solutions
3. Apply normalization techniques to make objectives comparable
4. Implement apporaches to find trade-off solutions
5. Make decisions from a Pareto frontier

Quick Recap: Local Search

Last week we optimized routes for delivery:

• Started with greedy construction (e.g. Nearest Neighbor)
• Improved with local search (e.g. 2-opt)
• Considered time windows
• But: We only optimized distance

Question: What if we also care about emissions, cost, AND customer satisfaction?

The Problem

Single vs Multi-Objective

Single Objective

• “Minimize total distance”
• Clear winner. Easy, right!

Multiple Objectives

• “Minimize cost AND emissions AND maximize speed”
• No clear answer…

Question: Any idea how to approach this?

EcoExpress Vehicle Options

Type	Purchase Cost (€)	Operating (€/km)	CO2 (g/km)	Speed (km/h)	Capacity (parcels)	Reliability
E-Truck	75000	0.18	0	55	300	0.93
Hybrid	45000	0.25	95	65	200	0.95
Diesel	35000	0.38	185	70	250	0.88
E-Bike	12000	0.05	0	30	50	0.98
Auto	95000	0.12	0	40	150	0.94

Question: Which vehicle is “best” for EcoExpress?

Trade-offs Everywhere

Every vehicle excels at something different!

Real Business Constraints

Beyond the numbers, consider:

• EU regulations: Carbon tax of €100/ton CO₂ starting 2025
• Competition: Amazon promises 2-hour delivery
• Labor market: Autonomous vehicles reduce driver dependency
• Urban zones: Zero-emission zones in city centers
• Peak times: Black Friday = 3x normal volume

There is no single “optimal” solution - only trade-offs

Pareto Optimality

Dominated Solutions

A solution is dominated if another solution is:

Better in at least one objective and not worse in any objective!

The Pareto Frontier

The Pareto frontier is the set of all non-dominated solutions

• No solution is objectively “better”
• Each represents a different trade-off
• Moving along frontier: gain in one objective, loss in another
• Decision makers choose based on preferences

Question Do you think you get the idea?

Find the Non-Dominated

Question: Which fleets are non-dominated?

Three+ Objectives

With 3 objectives, the Pareto frontier becomes a surface:

Harder to visualize, but same principle applies!

Fleet Composition

The Fleet Challenge

EcoExpress needs to replace their 80 diesel vans

• Must meet EU regulation: Average emissions ≤ 111 g CO₂/km
• Need capacity for 22,000 parcels/day
• Must balance cost vs. service quality
• 5 vehicle types available, each with trade-offs

Question: How do we choose the right mix?

Vehicle Options Recap

Type	Purchase Cost (€)	Operating (€/km)	CO2 (g/km)	Speed (km/h)	Capacity (parcels)	Reliability
E-Truck	75000	0.18	0	55	300	0.93
Hybrid	45000	0.25	95	65	200	0.95
Diesel	35000	0.38	185	70	250	0.88
E-Bike	12000	0.05	0	30	50	0.98
Auto	95000	0.12	0	40	150	0.94

Notice: No single vehicle is “best” at everything!

Fleet Composition Framework

This is a discrete selection problem, not continuous allocation

Decision Variables:

• Fleet: How many of each vehicle type? (discrete/integer)
• \(n_i\) = number of vehicles of type \(i\) (integers!)
• Example: \(n_{\text{E-Truck}} = 20\), \(n_{\text{Hybrid}} = 30\), etc.

Objective 1: Total Cost

Purchase cost + Operating cost over 3 years

\[\text{Total Cost} = \sum_{i} n_i \cdot \left( P_i + O_i \cdot d \cdot y \right)\]

• \(n_i\) = quantity of vehicle type \(i\)
• \(P_i\) = purchase cost of vehicle type \(i\)
• \(O_i\) = operating cost per km for type \(i\)
• \(d\) = daily distance × days per year
• \(y\) = years

Objective 2: Service Score

Composite measure of fleet performance

\[\text{Service Score} = 0.5 \cdot C_{\text{score}} + 0.3 \cdot R_{\text{score}} + 0.2 \cdot S_{\text{score}}\]

• \(C_{\text{score}} = \min\left(1.0, \frac{\text{Total Capacity}}{22000}\right)\) (capacity adequacy)
• \(R_{\text{score}} = \frac{\sum n_i \cdot r_i}{\sum n_i}\) (weighted avg. reliability)
• \(S_{\text{score}} = \frac{\sum n_i \cdot s_i}{70 \cdot \sum n_i}\) (normalized speed)

Service score captures multiple performance dimensions in one metric!

Hard Constraint: Emissions

EU regulation creates a feasibility boundary

\[\text{Average CO}_2 = \frac{\sum_{i} n_i \cdot e_i}{\sum_{i} n_i} \leq 111 \text{ g/km}\]

Where \(e_i\) = CO₂ emissions per km for vehicle type \(i\)

This eliminates some solutions:

• All diesel vans: 185 g/km > 111
• Mix with too many diesel: Still violates
• Zero-emission + some diesel: Might work

Data Source

Where Do These Numbers Come From?

Vehicle Specifications:

• Purchase costs: Manufacturer quotes, market research
• Operating costs: Fuel/electricity prices, maintenance records
• Capacity: Vehicle specs (cargo volume, weight limits)
• Reliability: Historical uptime data, manufacturer warranties
• EU Standards: WLTP certification for vehicles
• Electric vehicles: Grid carbon intensity (kWh → g CO₂)

Example Fleet Comparison

Three Fleet Strategies:

         name    cost  service        co2  capacity  vehicles
 Cost-Focused 28.9996 0.809705 120.714286     15000        70
     Balanced 19.0478 0.731840  33.928571     13250        70
Green-Focused 15.3102 0.695373   0.000000     12750        75


Cost-Focused: ✗ VIOLATES (CO2: 120.7 g/km)
Balanced: ✓ Compliant (CO2: 33.9 g/km)
Green-Focused: ✓ Compliant (CO2: 0.0 g/km)

Question: Which strategy would you choose?

Visualizing Fleet Trade-offs

Generated 68 feasible fleet compositions

Each point is a different fleet mix, all meeting emissions constraint!

Solution Approaches

Multi-Objective Optimization

You can use optimization solvers or heuristics!

With Optimization Solvers

• Weighted Sum Method
• ε-Constraint Method
  
• Goal Programming
• Optimal solutions
• Need mathematical model

With Heuristics

• Weighted Greedy Construction
• Multi-Objective Local Search
• Metaheuristics
• Good solutions, fast
• No optimality proof

In this lecture we use heuristic approaches!

Foundation: Extreme Points

First step for BOTH approaches - find the boundaries:

Question: Why is normalization essential?

Critical: Normalization

Without it, your analysis is meaningless

Question: Any intuition on how to do [0,1] normalization?

How to Normalize

The Normalization Formula for [0,1]

\[\text{Normalized}_i = \frac{x_i - x_{min}}{x_{max} - x_{min}}\]

In Python, this is rather simple!

def normalize_objectives(data):
    return (data - data.min()) / (data.max() - data.min())

# Now weights actually mean something
weighted_score = w1 * normalize(cost) + w2 * normalize(emissions)

Easy, right?

Extreme Points

There are several reasons why extreme points matter:

1. Trade-off Space: Min/max values bound your Pareto frontier
2. Enable Proper Normalization: Need ranges for scaling to [0,1]
3. Feasibility: If single objectives not achievable, problem infeasible
4. Stakeholder: “Best cost is €50k, best emissions is 40kg”

Implementation Pattern:

def find_extreme_points(problem):
    # Solve for minimum cost (ignore emissions)
    min_cost_solution = minimize(cost_objective, constraints)
    # Solve for minimum emissions (ignore cost)
    min_emissions_solution = minimize(emissions_objective, constraints)

Computational Complexity

How hard does it get with more objectives?

Why? Because there are just way more potential solutions to check!

Solver-Based Methods

Quick overview - you won’t implement these in assignments

1. Weighted Sum: Minimize \(w_1 \times \text{cost} + w_2 \times \text{emissions}\)
   • Simple, fast for convex problems
2. ε-Constraint: Minimize cost subject to emissions \(\leq \varepsilon\)
   • Systematically vary \(\varepsilon\) to find complete frontier
3. Goal Programming: Minimize deviations from targets
   • Set target for each objective, minimize weighted deviations

For your fleet optimization: You’ll use heuristic approaches instead!

Heuristic Approach

The Heuristic Strategy

For problems without mathematical models

1. Construction: Build initial solutions with weighted greedy
2. Improvement: Multi-objective local search
3. Selection: Filter dominated solutions to find Pareto frontier

Key difference from solvers:

• Solvers: Need mathematical model, guarantee optimality
• Heuristics: Work with any evaluation function, find good solutions fast

Why Heuristics?

Depending on the problem:

• Combinatorial explosion
• Huge solution space even for one problem
• Evaluating one solution might thus take too long
• Need diverse Pareto frontier, not just one “optimal” solution
• Open Source Solvers too slow
• Commercial solvers too expensive

Question: How do we build good solutions without a solver?

The Three-Stage Heuristic Process

This is what you’ll implement in your assignments!

Construct & Improve

Construction Methods for MOO

How to build initial solutions when you have multiple objectives?

Three choices (for starters). Let’s check them out!

Weighted Greedy Construction

Making greedy choices on a weighted objective

1. Choose weight vector w = (w₁, w₂)
2. At each step, pick the choice that minimizes: \[w_1 \cdot \text{cost}(x) + w_2 \cdot \text{emissions}(x)\]
3. Build complete solution greedily
4. Repeat with different weights to explore frontier

Different weights explore different trade-offs! Easy, right?

Sequential Greedy (Lexicographic)

Optimize one objective at a time, in priority order

1. Rank objectives by priority
   • E.g. cost (most important) and then emissions (tie-breaker)
2. At each step:
   • Find choices that minimize primary objective
   • If tie → use secondary objective
3. Build one working solution

We could also accept primary values within 10% of best so secondary has more influence!

Diverse Starting Pool

Generate many random solutions, keep the non-dominated ones

1. Generate N random solutions (e.g., N=100)
2. Evaluate all solutions on both objectives
3. Filter to keep only non-dominated solutions
4. Result: A diverse set of Pareto-optimal solutions

• Explores entire solution space
• No bias toward specific weights
• Great for warm-starting local search

Local Search for Multi-Objective

Special moves that improve multiple objectives:

Question: Which moves are acceptable?

MOO Local Search Rules

Accept a move if:

1. Dominance: New solution dominates current (win-win!)
2. Trade-off: Improves primary, acceptable loss in secondary
3. Probabilistic: Use temperature (like simulated annealing)

Always keep all your objectives in mind when making decisions.

Pareto to Decision

How to Choose!

1. The Knee Point: Find the “elbow” where improvement slows
2. Satisficing Levels: Set minimum acceptable thresholds
   • Cost must be < €100k (budget constraint)
   • Emissions must be < 100 kg (regulatory limit)
   • Service level must be > 90% (customer requirement)
3. Stakeholder Preferences: Let business priorities guide
   • Sustainability: Minimum emissions that meets constraints
   • Operations: Maximum service level within budget

Weighting has an Impact

The weights thus reflect your values!

Depending on your weight, the choice will vary.

Advanced

Speed vs Sustainability Dilemma

The Three-Way Trade-off in E-Commerce

1. Minimize Delivery Time (1-day/2-hour promise)
2. Minimize Cost (fuel, labor, fulfillment)
3. Minimize Environmental Impact (carbon footprint)

Faster delivery = More vehicles less full = Higher emissions

Question: What could retailers do?

Moving the Frontier

Instead of point on the frontier, move the entire frontier:

Question: Any idea of examples?

R&D can fundamentally change what’s possible!

Briefing

Today

Hour 2: This Lecture

• Multi-objective
• Pareto optimality
• Weighted greedy
• Local search MOO

Hour 3: Notebook

• Bean Counter CEO
• Find Pareto frontier
• Apply weighted greedy
• Normalize objectives

Hour 4: Competition

• Fleet composition
• Vehicle selection
• Cost vs service
• Justify choice!

The Competition Challenge

EcoExpress Sustainable Fleet Design

1. Select optimal fleet mix (5 vehicle types)
2. Balance cost vs. service score
3. Meet EU emission constraint (≤ 111 g CO₂/km)
4. Ensure sufficient capacity (22,000 parcels/day)

Find the best trade-off for your business priorities!

Choosing Your MOO Approach

Different situations call for different methods:

Situation	Best	Why
Clear priorities	Sequential greedy	Fast, hierarchy
Exploring	Weighted greedy	Different solutions
Many solutions	Diverse pool	Builds frontier
Quick solution	Single weighted	One good compromise
Improve existing	Multi-objective local	Refines trade-offs

Competition? Generate diverse pool or weigted, then improve with local search.

Implementation Pitfalls to Avoid

Common bugs that cost you time:

1. Forgetting to Normalize
   • Always normalize to [0,1] first!
2. Optimizing Too Many Objectives
   • 2-3: Manageable, 4+: Exponentially harder
   • Combine related objectives or use constraints
3. Not Checking Solution Feasibility
   • Always verify constraints after optimization

Summary

Key Takeaways:

• Real decisions have multiple conflicting objectives
• Pareto frontier shows all rational trade-offs
• Normalization is essential for fair comparison
• Weights reflect values, make them explicit
• Visualization crucial for decision-making

Break!

Take 20 minutes, then we start the practice notebook

Next up: You’ll become Bean Counter’s expert

Then: The Sustainability competition