## Objective function

### Total objective function

The total objective function $\textrm{LL}$ to be maximized is $\textrm{LL} = \sum_{i=1}^9\Lambda_i + \sum \textrm{pr} + \sum_y\sum_f\textrm{pen}_{y,f},$

including the log likelihoods $\Lambda_i$, logarithm of the parameter priors $pr$, and penalty functions, described below.

### Likelihoods

If the model is conditioned on catch and fishing mortality rates are estimated parameters, then the log-likelihood component $\Lambda_1$ of the catch is $\Lambda_1 = \sum_f \left[\lambda^{Y}_f \sum_y \left(-\log(\omega_{y,f}) - \dfrac{[\log(Y^{\textrm{obs}}_{y,f}) - \log(Y^{\textrm{pred}}_{y,f})]^2}{2 \omega _{y,f}^2}\right)\right],$

where $\textrm{obs}$ and $\textrm{pred}$ indicate observed and predicted quantities, respectively, $\lambda$ are likelihood weights, and $\omega$ is the catch standard deviation. With a small standard deviation for the catch likelihood relative to the variance in other likelihood components, i.e., $\omega = 0.01$, the predicted catch should match the observed catch in the model.

The log-likelihood component $\Lambda_2$ of survey data is $\Lambda_2 = \sum_s \left[ \lambda^I_s \sum_y \left(-\log(\sigma_{y,s}) - \dfrac{[\log(I^{\textrm{obs}}_{y,s}) - \log(I^{\textrm{pred}}_{y,s})]^2}{2\sigma_{y,s}^2}\right) \right].$

The log-likelihood component $\Lambda_3$ of catch-at-age data is $\Lambda_3 = \sum_f \lambda^A_f \left[\sum_y O^A_{y,f} \sum_a p^{\textrm{obs}}_{y,a,f} \log(p^{\textrm{pred}}_{y,a,f})\right],$ where $O^A$ is the annual sample sizes for the age compositions.

The log-likelihood component $\Lambda_4$ of catch-at-length data is $\Lambda_4 = \sum_f \lambda^L_f \left[ \sum_y O^L_{y,f} \sum_{\ell} p^{\textrm{obs}}_{y,\ell,f} \log(p^{\textrm{pred}}_{y,\ell,f})\right]$ where $O^L$ is the annual sample sizes for the length compositions.

The log-likelihood component $\Lambda_5$ for the observed mean size in this catch is:

$\Lambda_5 = \sum_f \lambda^{\bar{L}}_f\left[ \sum_y \left(-\log(\eta_{y,f}) - \dfrac{[\bar{L}^{\textrm{obs}}_{y,f} - \bar{L}^{\textrm{pred}}_{y,f}]^2}{2 \eta^2_{y,f}}\right)\right],$ for mean lengths, or

$\Lambda_5 = \sum_f \lambda^{\bar{w}}_f\left[ \sum_y \left(-\log(\eta_{y,f}) - \dfrac{[\bar{w}^{\textrm{obs}}_{y,f} - \bar{w}^{\textrm{pred}}_{y,f}]^2}{2 \eta^2_{y,f}}\right)\right],$ for mean weights, where $\eta_{y,f}$ is the standard deviation of the mean size. With constant coefficient of variation (CV), $\eta_{y,f} = \bar{w}^{\textrm{obs}}_{y,f} CV^{\eta}_f$.

The log-likelihood component $\Lambda_6$ of annual estimated recruitment deviates $\delta_y$ in log space is $\Lambda_6 = \Sigma_y\left(-\log(\tau) - \dfrac{\delta_y^2}{2 \tau^2}\right),$ where $\tau$ is the standard deviation of recruitment deviates.

The log-likelihood component $\Lambda_7$ of the equilibrium catch is $\Lambda_7 = \sum_f \lambda^{Y}_f \left(-\log(\omega^{\textrm{eq}}_f) - \dfrac{[\log(Y^{\textrm{eq,obs}}_f) - \log(Y^{\textrm{eq,pred}}_f)]^2}{2 \times (\omega^{\textrm{eq}}_f)^2}\right),$

The log-likelihood component $\Lambda_8$ of the survey proportion-at-age data is $\Lambda_8 = \sum_s \lambda^{IA}_s \left[\sum_y O^{IA}_{y,s} \sum_a p^{\textrm{obs}}_{y,a,s} \log(p^{\textrm{pred}}_{y,a,s})\right],$ where $O^{IA}$ is the annual sample sizes for the survey age compositions.

The log-likelihood component $\Lambda_9$ of the survey proportion-at-length data is $\Lambda_9 = \sum_s \lambda^{IL}_s \left[ \sum_y O^{IL}_{y,s} \sum_{\ell} p^{\textrm{obs}}_{y,\ell,s} \log(p^{\textrm{pred}}_{y,\ell,s})\right]$ where $O^{IL}$ is the annual sample sizes for the survey length compositions.

### Priors

Vague priors are added to selectivity parameters to aid in convergence, with \begin{align} x^{\textrm{LFS}}_b &\sim N(0,3)\\ x^{\textrm{L5}}_b &\sim N(0,3)\\ V^{L_{\infty}}_b &\sim \textrm{Beta}(1.01, 1.01) \end{align}

If free selectivity parameters are estimated, then $v_{y,a,f} \sim \textrm{Beta}(1.01, 1.01)$

### Optional priors

See sub-article on priors.

### Penalties

A penalty to the likelihood is added when $F_{y,f} \ge F_{\textrm{max}}$ for any year and fleet. The penalty is

$\textrm{pen}_{y,f} = \begin{cases} 0.01 (F_{y,f} - F_{\textrm{max}})^2 & F_{y,f} \ge F_{\textrm{max}}\\ 0 & \textrm{otherwise} \end{cases}.$