4. Dynamism¶

4.1 What Dynamism Means¶

Intuition

Dynamism means decisions are connected through time. You choose now, the system changes, new information arrives, and you choose again.

\[ \text{decision}\rightarrow\text{information}\rightarrow \text{decision}\rightarrow\text{information}\rightarrow\text{decision} \]

Formal definition

At decision point \(k\):

\[ S_k \rightarrow x \rightarrow S_k^x \rightarrow \omega_{k+1} \rightarrow S_{k+1} \]

Properties

Current decisions affect future states.
Future information may change what decision is best later.
A dynamic solution is a policy.
Timing matters: information cannot be used before it arrives.

sequenceDiagram
    participant M as Manager
    participant Sys as System
    participant W as World
    M->>Sys: observe S_k
    M->>Sys: choose x in X(S_k)
    Sys->>Sys: update to S_k^x
    W->>Sys: reveal omega_{k+1}
    Sys->>M: observe S_{k+1}

4.2 Exogenous and Endogenous Processes¶

Intuition

Exogenous means outside your control. Endogenous means caused by your decisions.

Formal distinction

Endogenous change: \(S_k\) and \(x\) determine the controlled part of \(S_k^x\).

Exogenous change: \(\omega_{k+1}\) determines the random part of \(S_{k+1}\).

Examples:

Exogenous	Endogenous
customer arrivals	accepting/rejecting customers
travel-time realization	route choice
weather	dispatching drivers
service duration	scheduling technicians
random container arrivals	shipping or holding containers

Worked example: pizza delivery

State:

current time
status of three delivery drivers
customers yet to be served

Decision:

send driver 1 to customer A

Endogenous effect:

driver 1 becomes busy; customer A is assigned

Exogenous information:

new order arrives; actual travel time is observed

Next state:

updated time, drivers, and open customer list

4.3 Policies and Adaptation¶

Intuition

A plan says "do these actions." A policy says "if this happens, do this; if that happens, do that."

Formal definition

\[ \pi:S\rightarrow X \]

Translation:

The policy maps each possible state to a decision.

Properties

A policy can be a rule, optimization model, learned value function, or look-ahead procedure.
The same policy can produce different actions in different states.
A good policy uses relevant information without becoming too complex to implement.

Worked example

Bad fixed plan:

Always dispatch the closest driver.

Better policy:

If demand is low, dispatch closest driver.
If downtown demand is forecast high, preserve at least one downtown driver.
If order is urgent, prioritize lateness reduction.

Exam phrase:

The dynamic problem should be solved by a policy because the decision maker needs to react to the state observed at each decision point.

4.4 Dynamic Timing: What Is Known When?¶

Intuition

Most mistakes in DDM modeling are timing mistakes. Students accidentally put future unknown demand into the current state, or they use information before it is revealed.

Before decision: state \(S_k\).
After decision: post-decision state \(S_k^x\).
After uncertainty: next state \(S_{k+1}\).

flowchart LR
    subgraph Before["Before decision"]
        A["Known now<br/>S_k"]:::state
    end
    subgraph Control["Controlled by decision"]
        B["Choose x"]:::decision
        C["Deterministic update<br/>S_k^x"]:::post
    end
    subgraph Random["Outside control"]
        D["Random information<br/>omega_{k+1}"]:::random
        E["New state<br/>S_{k+1}"]:::state
    end
    A --> B --> C --> D --> E
    classDef state fill:#e0f2fe,stroke:#0284c7,stroke-width:2px,color:#0f172a;
    classDef decision fill:#ffedd5,stroke:#f97316,stroke-width:2px,color:#0f172a;
    classDef post fill:#f3e8ff,stroke:#9333ea,stroke-width:2px,color:#0f172a;
    classDef random fill:#fee2e2,stroke:#dc2626,stroke-width:2px,color:#0f172a;

Worked example: meal delivery timing

At 6:00 PM:

\(S_k\) includes known open orders, current drivers, current restaurant prep status, and current time.

Decision:

\(x=\text{send driver A to restaurant R now}\).

Post-decision state:

\(S_k^x\) records that driver A is committed and en route.

New information:

\(\omega_{k+1}=\) a new order arrives, actual prep time is updated, and travel time is realized.

Next state:

\(S_{k+1}\) includes the updated driver position and the new order.

Approach comparison: replanning frequency

Replanning style	Merit	Disadvantage
fixed interval	predictable system load	may react slowly
event-based	reacts quickly to important events	irregular and complex
hybrid	practical balance	needs trigger rules

4.5 Slide Case: Static Planning vs Dynamic Replanning¶

Mental model

The dynamism lecture asks a simple operational question: if the world changes during the day, should we keep following the original plan or replan? Static planning is useful when change is small or replanning is impossible. Dynamic planning is valuable when new information is important and decisions can still be adapted.

Formal distinction

Static plan:

\[ (x_0,x_1,\ldots,x_K) \text{ chosen before uncertainty is revealed.} \]

Dynamic policy:

\[ X_k^\pi(S_k) \text{ chosen after observing } S_k. \]

Translation: a static plan commits to actions; a policy reacts to the latest state.

Worked example: urban delivery and traffic management

A city can change traffic strategies around pollution hotspots. The delivery company sees changed speeds and travel times. A static route may keep driving through a newly congested or restricted area. A dynamic policy recomputes the next customer or path using the updated state.

Feature	Static routing	Dynamic replanning
Traffic changes	ignored after plan is made	incorporated when observed
Decision object	complete route	next action or updated route
Main risk	plan becomes outdated	replanning overhead
Best when	system is stable	new information has high value

DDM interpretation

The exogenous process is the changing travel-time information. The endogenous process is the route or dispatch decision. DDM matters because these two processes interact over time.

4.6 Course Diagram: Exogenous and Endogenous Loop¶

flowchart LR
    subgraph Info["Information model / exogenous process"]
        A["Traffic, demand, service times"]:::random --> B["New information omega_{k+1}"]:::random
    end
    subgraph Decision["Decision model / endogenous process"]
        C["State S_k"]:::state --> D["Decision x in X(S_k)"]:::decision --> E["Post-decision state S_k^x"]:::post
    end
    B --> C
    E --> A
    classDef state fill:#dbeafe,stroke:#2563eb,stroke-width:2px,color:#0f172a;
    classDef decision fill:#ffedd5,stroke:#f97316,stroke-width:2px,color:#0f172a;
    classDef post fill:#ede9fe,stroke:#7c3aed,stroke-width:2px,color:#0f172a;
    classDef random fill:#fee2e2,stroke:#dc2626,stroke-width:2px,color:#0f172a;

Exam phrasing

The problem is dynamic because decisions occur repeatedly and each decision changes the state for later decisions. It is stochastic if new information such as travel time or demand is unknown when the current decision is made.

Chapter source note

Course basis: DDM dynamism slides, exogenous/endogenous process discussion, and pizza delivery tutorial state. Textbook enrichment: Powell's decision-information-decision framing for sequential decisions. In exam answers, make the timing of \(S_k\), \(x\), \(S_k^x\), \(\omega_{k+1}\), and \(S_{k+1}\) explicit.