Multi-Objective Optimization

Lecture 8 - Management Science

Author

Dr. Tobias Vlćek

Introduction

Client Briefing: EcoExpress Logistics

. . .

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:

Explain why most decisions involve competing objectives
Identify and visualize Pareto optimal solutions
Apply normalization techniques to make objectives comparable
Implement apporaches to find trade-off solutions
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 Problem

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:

Trade-off Space: Min/max values bound your Pareto frontier
Enable Proper Normalization: Need ranges for scaling to [0,1]
Feasibility: If single objectives not achievable, problem infeasible
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

Weighted Sum: Minimize \(w_1 \times \text{cost} + w_2 \times \text{emissions}\)
- Simple, fast for convex problems
ε-Constraint: Minimize cost subject to emissions \(\leq \varepsilon\)
- Systematically vary \(\varepsilon\) to find complete frontier
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

. . .

Construction: Build initial solutions with weighted greedy
Improvement: Multi-objective local search
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!

Construction & Improvement

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

Choose weight vector w = (w₁, w₂)
At each step, pick the choice that minimizes: \[w_1 \cdot \text{cost}(x) + w_2 \cdot \text{emissions}(x)\]
Build complete solution greedily
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

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

Generate N random solutions (e.g., N=100)
Evaluate all solutions on both objectives
Filter to keep only non-dominated solutions
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:

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

. . .

Always keep all your objectives in mind when making decisions.

From Pareto Front to Decision

How to Choose!

The Knee Point: Find the “elbow” where improvement slows
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)
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

Minimize Delivery Time (1-day/2-hour promise)
Minimize Cost (fuel, labor, fulfillment)
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

. . .

Select optimal fleet mix (5 vehicle types)
Balance cost vs. service score
Meet EU emission constraint (≤ 111 g CO₂/km)
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:

Forgetting to Normalize
- Always normalize to [0,1] first!
Optimizing Too Many Objectives
- 2-3: Manageable, 4+: Exponentially harder
- Combine related objectives or use constraints
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