The Objective Function of the Fleet

Objective function

We define the objective for a single focal fleet $f$.

Indices

• $f$: fleet
  
• $l$: patch (location)
  
• $s$: species
  
• $a$: age

Effort and fishing mortality

Let $E_{l,f}$ denote the effort of fleet $f$ in patch $l$. Fleet-specific fishing mortality at age is

$$F_{l,s,a,f} = q_{l,s,f}\,\text{sel}_{a,s,f}\,E_{l,f},$$

where $q_{l,s,f}$ is catchability and $\text{sel}_{a,s,f}$ is selectivity at age.

Let $F^{\text{ext}}_{l,s,a}$ denote exogenous fishing mortality from all fleets other than $f$, and let $m_{a,s}$ denote natural mortality. Total instantaneous mortality is

$$Z_{l,s,a} = F_{l,s,a,f} + F^{\text{ext}}_{l,s,a} + m_{a,s}.$$

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Revenue (Baranov catch)

Revenue for fleet $f$ is given by Baranov catch multiplied by price:

$$\text{Rev}_f = \sum_{l}\sum_{s}\sum_{a} \text{price}_{s,f} \; \frac{F_{l,s,a,f}}{Z_{l,s,a}} \left(1 - e^{-\Delta_t Z_{l,s,a}}\right) b_{l,s,a},$$

where $b_{l,s,a}$ is biomass and $\Delta_t$ is the time step.

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Cost (Normalized Formulation)

Parameters

• $c0_f$: cost-per-unit-effort scalar (calibrated by tune_fleets)
• $E^{\text{ref}}_f$: reference effort level (mean per-patch effort at equilibrium)
• $\gamma_f \ge 1$: effort cost exponent (congestion parameter; default 1.2)
• $\theta_f \ge 0$: travel weight, controls relative importance of spatial costs
• $\tilde{d}_{l,f}$: normalized cost per patch (spatial shape with mean 1 across open patches)

User-facing parameters

Users specify the following interpretable parameters in create_fleet:

Parameter	Meaning	Default
cr_ratio	Cost/revenue ratio at equilibrium	1
effort_cost_exponent ($\gamma$)	Convexity of congestion costs	1.2
travel_fraction	Share of costs from travel at equilibrium	0
cost_per_patch	Spatial cost pattern (normalized internally)	uniform

The travel weight $\theta_f$ is derived from travel_fraction as:

$$\theta_f = \frac{\text{travel\_fraction}}{1 - \text{travel\_fraction}}$$

This parameterization ensures that travel costs constitute exactly travel_fraction of total costs at equilibrium when effort is uniformly distributed across open patches.

Cost formula

Total cost for fleet $f$ is:

$$\text{Cost}_f = c0_f \cdot E^{\text{ref}}_f \left( \sum_{l} \left(\frac{E_{l,f}}{E^{\text{ref}}_f}\right)^{\gamma_f} + \theta_f \sum_{l} \tilde{d}_{l,f} \cdot \frac{E_{l,f}}{E^{\text{ref}}_f} \right)$$

Key properties:

1. Effort normalization: Dividing by $E^{\text{ref}}_f$ makes costs scale-invariant. Changing grid resolution or base_effort doesn’t affect the relative importance of congestion vs. travel.

2. Separable components:

   • Convex term $\sum_l (E_l / E^{\text{ref}})^{\gamma}$: congestion/diminishing returns
   • Travel term $\theta \sum_l \tilde{d}_l (E_l / E^{\text{ref}})$: spatial heterogeneity

3. Interpretable parameters:

   • $\gamma$ controls congestion severity (1 = linear, >1 = convex)
   • $\theta$ (via travel_fraction) controls how much spatial location matters
   • $\tilde{d}_l$ is a pure spatial shape (mean 1), making travel_fraction directly interpretable

Tuning

The tune_fleets function calibrates $c0_f$ to achieve the target cost/revenue ratio:

$$c0_f = \frac{\text{cr\_ratio} \cdot \text{Rev}_f}{E^{\text{ref}}_f \cdot P_f \cdot (1 + \theta_f)}$$

where $P_f$ is the number of open fishing patches. The function also computes and stores the equilibrium $E^{\text{ref}}_f$ as the mean per-patch effort.

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Final objective

The objective for fleet $f$ is

$$\begin{aligned} \max_{\{E_{l,f}\}} \;\; \text{obj}_f &= \sum_{l}\sum_{s}\sum_{a} \text{price}_{s,f} \; \frac{q_{l,s,f}\,\text{sel}_{a,s,f}\,E_{l,f}} {q_{l,s,f}\,\text{sel}_{a,s,f}\,E_{l,f} + F^{\text{ext}}_{l,s,a} + m_{a,s}} \\[6pt] &\quad \times \left(1 - e^{-\Delta_t\left( q_{l,s,f}\,\text{sel}_{a,s,f}\,E_{l,f} + F^{\text{ext}}_{l,s,a} + m_{a,s} \right)}\right) b_{l,s,a} \\[10pt] &\quad - c0_f \cdot E^{\text{ref}}_f \left( \sum_{l} \left(\frac{E_{l,f}}{E^{\text{ref}}_f}\right)^{\gamma_f} + \theta_f \sum_{l} \tilde{d}_{l,f} \cdot \frac{E_{l,f}}{E^{\text{ref}}_f} \right). \end{aligned}$$

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Constraints

The optimization is subject to:

• Total effort constraint: $$\sum_l E_{l,f} = E^{\text{tot}}_f$$
• Non-negativity: $$E_{l,f} \ge 0$$
• Fleet-specific spatial closures, enforced by setting $E_{l,f} = 0$ in closed patches.