# 37 Constrained optimization of fishing effort

Optimization methods like Ecosim’s policy optimization search procedure (see Fishing policy exploration chapter) have been used to find, by fleet, fishing efforts that maximize various multi criteria benefit functions for ecosystem management, with criteria ranging from total profits to total employment and maintenance of ecosystem structure. These optimization methods work by running an ecosystem model that includes fishing mortality rates for multiple biomass groups by the multiple fleets, for long enough simulation periods for production and trophic interaction effects to play out. Fishing efforts by fleet are varied across simulation runs so as to seek the effort combination that maximizes the long-term benefit function. The result is a vector *EOPT _{j}*,

*j=1,…,nf*, of optimum long-term efforts for the

*nf*fleets included in the model.

A relatively simple linear programming method could theoretically be used to find long-term efforts that are constrained to be at or below the long term optima EOPT* _{j}* found in the policy optimization, while respecting the relative fishery values represented in the long term optimization and also constraints associated with fishing mortality rate targets or upper limits by species. We have not attempted to implement that theory. But the same optimization procedure can also be used to set annual effort limits in MSE simulations, as an option to use effort input controls rather than the output (quota) controls assumed in an older EwE module and the newer CEFAS MSE module (see MSE chapter). The present chapter explains the basic linear programming formulation.

Unfortunately, simple long term optimum long-term efforts typically cannot be used directly in management strategy evaluations (MSEs), since they ignore constraints associated with fishery development (how fast efforts can grow or be reduced) and more importantly typically involve fishing efforts that would cause ecosystem simplification (overfishing of weaker stocks, even use of fleets to cull some species so as to increase productivity of others). In at least some jurisdictions like the United States, there is a strict legal mandate prohibiting over-harvesting (fishing rate exceeding *F _{msy}*) for any species. Further, in modeling management decision making over time, it is often necessary to represent the use of “control rules” that prescribe reduced fishing mortality rates for particular stocks when those stocks are below desired reference biomass levels.

A relatively simple linear programming method suggested by Murawksi and Finn^{[1]} can be used to find short-term (annual) efforts that are constrained to be at or below the long-term optima *EOPT _{j},* while respecting the relative fishery values represented in the long-term optimization and also constraints associated with fishing mortality rate targets or upper limits by species. Suppose that the fleets harvest

*i=1,…,ns*species (or biomass groups), with each unit of effort for fleet

*j*causing a fishing mortality rate

*q*on species

_{ij}*i*(

*q*is the catchability coefficient for species

_{ij}*i*by fleet

*j*, and is also the Ecopath base fishing mortality rate for

*i,j*). Suppose that for each species

*i*, there is a target or maximum allowable fishing mortality rate

*FTARGET*

_{i}, summed over all fleets that cause fishing mortality (landings and/or discards) on

*i*. Suppose that we assign an “importance weight”

*v*to effort by fleet

_{j}*j*, where

*v*reflects the relative value of increasing (or maintaining) effort for fleet

_{j}*j*because of its contribution to overall long term ecosystem value and/or its legal entitlement to fish. Then the linear programming optimization can be formulated simply, as, maximize,

[latex]\sum \limits _{j=1}^{nf} v_j E_j \tag{1}[/latex]

by varying the efforts *E _{j }*subject to the constraints,

[latex]EMIN_j < E_j < EOPT_j \text{ for j=1, ..., nf} \tag{2}[/latex]
[latex]\sum\limits_{j=1}^{nf}q_{ij} E_j \leq FTARGET_i \text{ for i=1, ..., ns} \tag{3}[/latex]
That is, try to make the efforts *E*_{j} as large as possible without exceeding *EOPT _{j}* while allowing efforts of at least

*EMIN*and not allowing the sums of

_{j }*q*to exceed target fishing rate

_{ij}E_{j}*FTARGET*for any species

_{i}*i*. An alternative formulation for recognizing fishery development rate constraints would be to replace the first set of

*nf*constraints (Eq. 2) with

*EMIN*. Where the

_{j}< E_{j}< EMAX_{j}*EMIN*and

*EMAX*are allowed to vary from year to year by limited increments from the previous year’s values, and are not allowed to exceed

*EOPT*.

Figure 1 illustrates the kind of complicated solutions that can arise from this optimization, even for a very simple case where two fishing fleets pursue just two species, with one fleet having higher catchability for one of the species and the other fleet having higher catchability for the other species.

**Figure 1. Graphical representation of the linear programming problem. Each line represents a constraint (vertical and horizontal lines are the EOPT constraints, sloped lines are the FTARGET constraints.**

The lines in Figure 1 represent effort levels that exactly meet the constraints; efforts must be to the left and below each line in order to be feasible. Thus the feasible effort combinations are only those in the polygon from the graph origin out to the first constraint lines met. Since efforts are to be as large as possible, the solution has to lie along one of those first lines met, and in fact has to be on one of the three vertices marked A, B, C.

- Effort combination A represents Fleet 1 being severely restricted, but Fleet 2 operating at its
*EOPT.* - Effort combination C represents Fleet 2 being severely restricted but Fleet 1 operating at its
*EOPT*, and - Effort combination B represents a “balanced” policy choice where both fleets are restricted to below
*EOPT*in order to “share the burden” of avoiding exceeding either of the two*FTARGET*species constraints.

Which of these three combinations will be chosen (solve the linear programming maximization) depends on the value weights *v _{j}* in Eq. 1. If

*v*is much higher than

_{1}*v*, combination C will be chosen, combination A will be chosen if

_{2}*v*is larger, and setting the two

_{2}*v*’s equal will more likely lead to the balanced combination B. Note also in Figure 1 that as the

*FTARGET*constraints are “relaxed” (increased so the sloped lines move upward and to the right), it becomes more likely that the optimum effort combination will lie near

*EOPT*for both fleets; likewise, as these constraints are “tightened” (reduced so the sloped lines move down and to the left), it becomes more likely that the

*EOPT*s will not be in the feasible region so that the optimum solution will lie either with a mixed effort combination or with one or another of the fleets shut down entirely.

Using the linear programming formulation, it is simple to evaluate the cost, in terms of lost total value, of introducing more restrictive constraints on species harvest rates. In the overall management strategy evaluation setup for Ecosim, the only other way to evaluate this cost is to do policy runs with and without “weakest stock” constraints on fleet quotas, where all fleets are assumed to share equally in reductions needed to meet such constraints. The linear programming solution may well demonstrate that such equal sharing of the conservation burden is in fact far from optimum.

The linear programming formulation can also be used to demonstrate potential increases in fishery value from selective fishing practices that change the species-specific catchabilities qij. For example, if *q _{12}* and

*q*(catchability of species 1 by fleet 2 and of species 2 by fleet 1) could be greatly reduced in the Figure 1 example, the slopes of the two

_{21}*FTARGET*constraint lines would decrease/increase so as to move the solution toward higher total efforts (move point B up and to the right, closer to the

*EOPT1-EOPT2*intersection) and thus higher total value.

The key to getting useful results from the linear programming exercise is to make wise choice of the fleet value weights *v _{j}*. One objective option for doing this each year (assuming a management strategy where biomass and perhaps catchability

*q*estimates are being updated regularly) is to set each weight to be

_{ij}[latex]v_j = \sum \limits _{i=1}^{nf}P_{ij}q_{ij}B_i \tag{4}[/latex]

where *P _{ij}* is the landed price for species

*i*by fleet

*j*and

*B*is the current estimated biomass of species

_{i}*i*. Using this formula,

*v*is just the sum over species of predicted catches per effort times prices, so that

_{j}*v*represents the (short term) predicted total value of landings by fleet

_{j}B_{j}*j*and the overall linear programming objective function just becomes the predicted total landed value of all catches.

An option in the management strategy evaluation interface allows users to replace the complicated older EwE and CEFAS rules for setting quotas to limit mortality rates to acceptable levels, to instead use annual LP optimization each year during each MSE simulation run so as to limit fishing efforts rather than quotas. The basic idea is just to replace the single E_{j} values in the equations above with time-varying values calculated for each year and group using annual values for the FTARGET_{i} target fishing rate for each group i for that year, calculated from the estimated biomass of the group and the group’s harvest control rule that specifies how target fishing rate for the group should vary with stock size. This option can lead to complex policy changes over time, especially when the *FTARGET _{i}* decrease with decreases in biomasses

*B*.

_{i}- Murawski, S.A. and J.T. Finn. 1986. Optimal effort allocation among competing mixed-species fisheries, subject to fishing mortality constraints. Can. J. Fish. Aquat. Sci. 43: 90-100. https://doi.org/10.1139/f86-010 ↵