{"id":3623,"date":"2024-05-02T13:23:33","date_gmt":"2024-05-02T17:23:33","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/?post_type=chapter&p=3623"},"modified":"2025-10-30T08:57:32","modified_gmt":"2025-10-30T12:57:32","slug":"constrained-optimization-of-fishing-effort","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/chapter\/constrained-optimization-of-fishing-effort\/","title":{"raw":"Constrained optimization of fishing effort","rendered":"Constrained optimization of fishing effort"},"content":{"raw":"

Optimization methods like Ecosim\u2019s policy optimization search procedure (see Fishing policy exploration<\/a> 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.\u00a0\u00a0 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.\u00a0 Fishing efforts by fleet are varied across simulation runs so as to seek the effort combination that maximizes the long-term benefit function.\u00a0 The result is a vector EOPTj<\/sub><\/em>, \u00a0j=1,\u2026,nf<\/em>, of optimum long-term efforts for the nf<\/em> fleets included in the model.<\/p>\r\nA 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 EOPTj<\/sub><\/em> 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<\/a>).\u00a0 The present chapter explains the basic linear programming formulation.\r\n

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).\u00a0 In at least some jurisdictions like the United States, there is a strict legal mandate prohibiting over-harvesting (fishing rate exceeding Fmsy<\/sub><\/em>) for any species.\u00a0 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.<\/p>\r\n

A relatively simple linear programming method suggested by Murawksi and Finn[footnote]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<\/a> [\/footnote] can be used to find short-term (annual) efforts that are constrained to be at or below the long-term optima EOPTj<\/sub>,<\/em> 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.\u00a0 Suppose that the fleets harvest i=1,\u2026,ns<\/em> species (or biomass groups), with each unit of effort for fleet j<\/em> causing a fishing mortality rate qij<\/sub><\/em> on species i<\/em> (qij<\/sub><\/em> is the catchability coefficient for species i<\/em> by fleet j<\/em>, and is also the Ecopath base fishing mortality rate for i,j<\/em>).\u00a0 Suppose that for each species i<\/em>, there is a target or maximum allowable fishing mortality rate FTARGET<\/em>i<\/sub>, summed over all fleets that cause fishing mortality (landings and\/or discards) on i<\/em>.\u00a0 Suppose that we assign an \"importance weight\" vj<\/sub><\/em> to effort by fleet j<\/em>, where vj<\/sub><\/em> reflects the relative value of increasing (or maintaining) effort for fleet j<\/em> because of its contribution to overall long term ecosystem value and\/or its legal entitlement to fish.\u00a0 Then the linear programming optimization can be formulated simply, as, maximize,<\/a><\/p>\r\n[latex]\\sum \\limits _{j=1}^{nf} v_j E_j \\tag{1}[\/latex]\r\n\r\nby varying the efforts Ej <\/sub><\/em>subject to the constraints,<\/a>\r\n\r\n[latex]EMIN_j < E_j < EOPT_j \\text{ for j=1, ..., nf} \\tag{2}[\/latex]\r\n\r\n[latex] \\sum\\limits_{j=1}^{nf}q_{ij} E_j \\leq FTARGET_i \\text{ for i=1, ..., ns} \\tag{3}[\/latex]\r\n\r\nThat is, try to make the efforts E<\/em>j<\/sub> as large as possible without exceeding EOPTj<\/sub><\/em>\u00a0while allowing efforts of at least EMINj <\/sub><\/em>and not allowing the sums of qij<\/sub>Ej<\/sub><\/em> to exceed target fishing rate FTARGETi<\/sub><\/em> for any species i<\/em>.\u00a0 An alternative formulation for recognizing fishery development rate constraints would be to replace the first set of nf<\/em> constraints (Eq. 2<\/a>) with EMINj<\/sub> < Ej<\/sub> < EMAXj<\/sub><\/em>. Where the EMIN<\/em> and EMAX<\/em> are allowed to vary from year to year by limited increments from the previous year\u2019s values, and are not allowed to exceed EOPT<\/em>.\r\n

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.<\/p>\r\n

\"\"<\/p>\r\n

Figure 1.\u00a0 Graphical representation of the linear programming problem.\u00a0 Each line represents a constraint (vertical and horizontal lines are the EOPT<\/em> constraints, sloped lines are the FTARGET<\/em> constraints.<\/strong><\/p>\r\n

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.\u00a0 Thus the feasible effort combinations are only those in the polygon from the graph origin out to the first constraint lines met.\u00a0 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.<\/p>\r\n\r\n