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**Music-Inspired Optimization Algorithm Harmony Search**

Zong Woo Geem

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What is Optimization? Procedure to make a system or design as effective, especially the mathematical techniques involved. ( Meta-Heuristics) Finding Best Solution Minimal Cost (Design) Minimal Error (Parameter Calibration) Maximal Profit (Management) Maximal Utility (Economics)

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**Types of Optimization Algorithms**

Mathematical Algorithms Simplex (LP), BFGS (NLP), B&B (DP) Drawbacks of Mathematical Algorithms LP: Too Ideal (All Linear Functions) NLP: Not for Discrete Var. or Complex Fn., Feasible Initial Vector, Local Optima DP: Exhaustive Enumeration, Wrong Direction Meta-Heuristic Algorithms GA, SA, TS, ACO, PSO, …

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**Existing Nature-Inspired Algorithms**

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**Existing Meta-Heuristic Algorithms**

Definition & Synonym Evolutionary, Soft computing, Stochastic Evolutionary Algorithm (Evolution) Simulated Annealing (Metal Annealing) Tabu Search (Animal’s Brain) Ant Algorithm (Ant’s Behavior) Particle Swarm (Flock Migration) Mimicking Natural or Behavioral Phenomena → Music Performance

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**Algorithm from Music Phenomenon**

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**Algorithm from Jazz Improvisation**

Click Below

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**Analogy = Do = Mi = Sol = 100mm = 300mm = 500mm f (100, 300, 500)**

Mi, Fa, Sol Do, Re, Mi Sol, La, Si = Do = Mi = Sol 100mm 200mm 300mm 300mm 400mm 500mm 500mm 600mm 700mm f (100, 300, 500) = 100mm = 300mm = 500mm

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**Comparison Factors Musical Inst. → Decision Var.**

Pitch Range → Value Range Harmony → Solution Vector Aesthetics → Objective Function Practice → Iteration Experience → Memory Matrix

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**Good Harmony & Bad Harmony**

An Algorithm which Keeps Better Harmonies!

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**Procedures of Harmony Search**

Step 0. Prepare a Harmony Memory. Step 1. Improvise a new Harmony with Experience (HM) or Randomness (rather than Gradient). Step 2. If the new Harmony is better, include it in Harmony Memory. Step 3. Repeat Step 1 and Step 2.

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**HS Operators Random Playing Memory Considering Pitch Adjusting**

Ensemble Considering Dissonance Considering

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Random Playing x ∈ Playable Range = {E3, F3, G3, A3, B3, C4, D4, E4, F4, G4, A4, B4, C5, D6, E6, F6, G6, A6, B6, C7}

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Memory Considering x ∈ Preferred Note = {C4, E4, C4, G4, C4}

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Pitch Adjusting x+ or x-, x ∈ Preferred Note

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Ensemble Considering

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**Rule Violation (Parallel 5th)**

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**Example of Harmony Search**

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**Initial Harmony Memory**

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Next Harmony Memory

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With Three Operators {1, 2, 3, 4, 5} +1 f = 6 1 4 2

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**HS Applications for Benchmark Problems**

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**Six-Hump Camel Back Function**

f*( , ) = (Exact) f ( , ) = (HS)

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Multi-Modal Function

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**Artificial Neural Network - XOR**

T F Bias Sum of Errors in BP = 0.010 Sum of Errors in HS = 0.003

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**HS Applications for Real-World Problems**

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Sudoku Puzzle 6 1 4 2 5 3 8 7 9

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**Music Composition – Medieval Organum**

Interval Rank Fourth 1 Fifth 2 Unison 3 Octave Third 4 Sixth Second 5 Seventh

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**Project Scheduling (TCTP)**

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**University Timetabling**

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Internet Routing

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**Web-Based Parameter Calibration**

RMSE: (Powell), (GA), (HS)

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**Truss Structure Design**

GA = , HS =

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**School Bus Routing Problem**

Depot School 1 2 3 4 5 6 7 8 9 10 15 20 Min C1 (# of Buses) + C2 (Travel Time) s.t. Time Window & Bus Capacity GA = $409,597, HS = $399,870

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**Generalized Orienteering Problem**

Max. Multi-Objectives 1. Natural Beauty 2. Historical Significance 3. Cultural Attraction 4. Business Opportunity Case1 Case2 Case3 Case4 Case5 ANN 12.38 13.05 12.51 12.78 12.36 HS 13.08 12.56 12.40

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**Water Distribution Network Design**

1 2 3 4 5 6 7 8 9 15 14 11 18 12 13 17 10 19 16 20 21 MP: $78.09M GA: $38.64M (800,000) SA: $38.80M (Unknown) TS: $37.13M (Unknown) Ant: $38.64M (7,014) SFLA: $38.80M (21,569) CE: $38.64M (70,000) HS: $38.64M (3,373) 5 times out of 20 runs

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**Large-Scale Water Network Design**

Huge Variables (454 Pipes) GA = 2.3M Euro HS = 1.9M Euro

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**Multiple Dam Operation**

Max. Benefit (Power, Irrigation) GA = 400.5, HS = (GO)

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**Hydrologic Parameter Calibration**

Wedge Storage = K x (I - O) Prism Storage = K O O I Mathematical = , GA = 38.23, HS = 36.78

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**Ecological Conservation**

With 24 Sites, SA = 425, HS = 426

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**Satellite Heat Pipe Design**

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**Satellite Heat Pipe Design**

BFGS HS Minimize Mass Maximize Conductance BFGS: Mass =25.9 kg, Conductance = W/K HS: Mass = 25.8 kg, Conductance = W/K

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**Oceanic Oil Structure Mooring**

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**RNA Structure Prediction**

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Medical Imaging

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Radiation Oncology

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**Astronomical Data Analysis**

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**All that Jazz Robotics Visual Tracking Internet Searching**

Management Science Et Cetera

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**Paradigm Shift a change in basic assumptions within the ruling theory of science**

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**Stochastic Partial Derivative of HS**

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**Stochastic Co-Derivative of HS**

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**Parameter-Setting-Free HS**

Overcoming Existing Drawbacks Suitable for Discrete Variables No Need for Gradient Information No Need for Feasible Initial Vector Better Chance to Find Global Optimum Drawbacks of Meta-Heuristic Algorithms Requirement of Algorithm Parameters

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**Wikipedia (Web Encyclopedia)**

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**Books on Harmony Search**

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**Visitor Clustering (As of Nov. 2010)**

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Citations in Major Literature in tantum ut si priora tua fuerint parva, et novissima tua multiplicentur nimis. Iob 8:7

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**What is Your Contribution?**

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**Question for Harmony Search?**

Visit the Website HarmonySearch.info

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