ppa-network

Calculations

Variables:

• $P(t)$: Price charged per parcel at time $t$.
• $D(t)$: Demand (number of parcels) at time $t$, calculated using the demand function.
• $C(t)$: Total cost at time $t$, including fixed and variable costs.
• $N(t)$: Number of active network components (e.g., warehouses, routes) at time $t$.
• $F$: Fixed cost associated with operating one network component for a unit of time.
• $V$: Variable cost incurred per parcel handled.
• $a$: Maximum potential demand when the price is zero; represents the intercept of the demand function (total market volume).
• $b$: Demand sensitivity to price; indicates how much demand decreases with a unit increase in price.
• $K$: Capacity limit per network component; maximum number of parcels a component can handle in a given time period.
• $\Pi$: Total profit over the time period $T$.
• $T$: The time horizon over which the optimization is performed.

Assumptions:

1. Demand Function:

   $D(t) = a - bP(t)$

   • ( a ): Represents the highest possible demand when the price is zero.
   • ( b ): Indicates how much demand decreases as price increases; it reflects the price elasticity of demand.

2. Cost Function:

   $C(t) = N(t) \times F + V \times D(t)$

   • $N(t) \times F$: Total fixed costs at time $t$, based on the number of active components.
   • $V \times D(t)$: Total variable costs at time $t$, dependent on the number of parcels handled.

3. Capacity of Each Network Component:

   • Each component has a capacity limit $K$, representing the maximum number of parcels it can handle per unit of time (e.g., per day).

Objective Function:

• Profit $\Pi$:

  $\Pi = \int_{T} [P(t) \times D(t) - C(t)] , dt$

  Where $T$ is the time period over which we are optimizing.

Constraints:

1. Demand Allocation Constraint:

   $D(t) \leq N(t) \times K$

   • Ensures that total demand does not exceed the total capacity provided by the active network components.

2. Network Component Activation:

   • $N(t)$ is an integer representing the number of active components and can be adjusted over time.

3. Non-Negative Variables:

   • $D(t) \geq 0$
   • $P(t) \geq 0$
   • $N(t) \geq 0$ and integer

Optimization Problem:

• Maximize $\Pi$ with respect to $P(t)$ and $N(t)$, subject to the constraints.

Solution Approach:

1. Determine Optimal Price $( P^*(t) )$:

   • Use the demand function to relate price and demand.
   • For a given $N(t)$, solve for $P(t)$ that maximizes profit while ensuring $D(t) \leq N(t) \times K$.

2. Determine Optimal Number of Active Components $( N^*(t) )$:

   • Based on forecasted demand $D(t)$, adjust $N(t)$ to minimize costs while meeting demand.

3. Iterative Optimization:

   • Iterate between adjusting $P(t)$ and $N(t)$ to find the combination that maximizes profit.