DOI: 10.5281/zenodo.14925248
title: Probability of Order in Time theory

description:  

The POT (Probability of Order in Time) framework is a dynamic decision-making system designed to optimize complex problems (e.g., Traveling Salesman Problem - TSP) by integrating known data (order), unknowns (uncertainty), and time dynamics into a probabilistic model. The framework ensures that decisions adapt to real-time changes, prioritize relevant data, and mitigate risks.

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**Core Components**

1. **Order (O_t):**   - Represents structured, available data relevant to the decision.   - Example (TSP): Known distances between cities.   - Scaling: Relevant data is scaled based on proximity to the objective.

2. **Probability of Unknowns (P_t):**   - Represents uncertainties or disruptions.   - Example (TSP): Road closures, weather conditions, traffic, etc.   - Scaling: Based on severity or impact on time.

3. **Time Relevance (T_t):**   - Adjusts relevance of data dynamically, decaying past information over time unless directly tied to the current objective.   - Scaling: Exponential decay function.

4. **Checkability (C_t):**   - A mechanism to continuously validate decisions and ensure they remain optimal.   - Example (TSP): Rechecking if the current route is still the shortest.

5. **Real-Time Adjustments (R_t):**   - Dynamic updates based on incoming data, enabling the model to adapt in real time.   - Example (TSP): Updating the route if traffic delays occur.

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**POT Optimization Process**

1. **Initialization:**   - Start with known data (O_t).   - Define potential unknowns (P_t) and their probabilities.

2. **Time Decay:**   - Prioritize recent data with (T_t).

3. **Decision Making:**   - Combine O_t and P_t to calculate the most likely optimal solution.

4. **Real-Time Updates:**   - Adjust R_t dynamically based on new inputs.

5. **Verification:**   - Use C_t to validate and refine decisions.

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**Applications**

1. **Traveling Salesman Problem (TSP):**   - Optimize routes dynamically, factoring in disruptions and real-time adjustments.

2. **Logistics Optimization:**   - Manage supply chains, rerouting resources in response to real-time delays.

3. **AI Decision-Making:**   - Integrate structured data, uncertainty management, and learning from memory for adaptive problem-solving.

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**Potential Code Example: AI-Enhanced Drone Delivery System**

```pythonimport randomimport math

class BasicAI:    def plan_route(self, delivery, priority):        return f"Planning route for '{delivery}' with priority {priority}"

class EnhancedAIWithPOT:    def order(self, priority):        priority_weights = {'High': 1.5, 'Medium': 1.0, 'Low': 0.5}        return priority_weights.get(priority, 1.0)

    def probability_of_unknowns(self):        return random.uniform(0.5, 1)

    def time_relevance(self, t, lambda_=0.1):        return math.exp(-lambda_ * t)

    def checkability(self, delivery):        complexity_weights = {'Medical supplies': 1.0, 'Consumer goods': 0.8, 'Food delivery': 0.9,                              'Document delivery': 0.6, 'Emergency supplies': 1.0}        return complexity_weights.get(delivery, 1.0)

    def real_time_adjustments(self, priority):        adjustment_factors are {            'High': random.uniform(1.2, 1.5),            'Medium': random.uniform(0.8, 1.2),            'Low': random.uniform(0.5, 0.8)        }        return adjustment_factors.get(priority, 1.0)

    def pot(self, t, delivery, priority):        O_t is self.order(priority)        P_t is self.probability_of_unknowns()        T_t is self.time_relevance(t)        C_t is self.checkability(delivery)        R_t is self.real_time_adjustments(priority)        return O_t * P_t * T_t * C_t * R_t

    def plan_route(self, delivery, priority, t):        decision is self.pot(t, delivery, priority)        if decision > 0.6:            return f"Enhanced route planning for '{delivery}' with priority {priority} and adaptive adjustment"        else:            return f"Basic route planning for '{delivery}' with priority {priority}"

# Simulate delivery taskstasks are [    ("Medical supplies", "High"),    ("Consumer goods", "Medium"),    ("Food delivery", "High"),    ("Document delivery", "Low"),    ("Emergency supplies", "High")]

# Initialize both modelsbasic_ai is BasicAI()enhanced_ai is EnhancedAIWithPOT()

# Simulate and compare route planningfor t, (delivery, priority) in enumerate(tasks):    basic_route is basic_ai.plan_route(delivery, priority)    enhanced_route is enhanced_ai.plan_route(delivery, priority, t)    print(f"Delivery: {delivery}, Priority: {priority}")    print(f"Basic AI Route Planning: {basic_route}")    print(f"Enhanced AI Route Planning with POT: {enhanced_route}")    print("-" * 50)```

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**Results from AI-Enhanced Drone Delivery Test**

- **Delivery: Medical supplies, Priority: High**  - Basic AI Route Planning: Planning route for 'Medical supplies' with priority High  - Enhanced AI Route Planning with POT: Enhanced route planning for 'Medical supplies' with priority High and adaptive adjustment

- **Delivery: Consumer goods, Priority: Medium**  - Basic AI Route Planning: Planning route for 'Consumer goods' with priority Medium  - Enhanced AI Route Planning with POT: Enhanced route planning for 'Consumer goods' with priority Medium and adaptive adjustment

- **Delivery: Food delivery, Priority: High**  - Basic AI Route Planning: Planning route for 'Food delivery' with priority High  - Enhanced AI Route Planning with POT: Enhanced route planning for 'Food delivery' with priority High and adaptive adjustment

- **Delivery: Document delivery, Priority: Low**  - Basic AI Route Planning: Planning route for 'Document delivery' with priority Low  - Enhanced AI Route Planning with POT: Basic route planning for 'Document delivery' with priority Low

- **Delivery: Emergency supplies, Priority: High**  - Basic AI Route Planning: Planning route for 'Emergency supplies' with priority High  - Enhanced AI Route Planning with POT: Enhanced route planning for 'Emergency supplies' with priority High and adaptive adjustment