A nonlinear optimization model for obtaining a total ...

Journal of Applied Operational Research (2013) 5(4), 153–163 © Tadbir Operational Research Group Ltd. All rights reserved. www.tadbir.ca ISSN 1735-8523 (Print), ISSN 1927-0089 (Online)

A nonlinear optimization model for obtaining a total allowable catch quota of the Chilean jack mackerel fishery Víctor M. Albornoz 1,* and Cristian M. Canales 2 1 2

Universidad Técnica Federico Santa María, Santiago, Chile Instituto de Fomento Pesquero, Valparaíso, Chile

Abstract. A methodology proposed for obtaining an optimal annual allowable capture quota of Chilean jack mackerel is described as a planning tool in the exploitation of this fishery resource. More precisely, a nonlinear optimization model i s formulated that considers as decision variables an annual capture quota for each period of the planning horizon, decisions related to the size of the populations in each period considering the age structure of the population and its exploitation zones. The proposed decisions are based mainly on the knowledge of the population dynamics and conditions that ensure the sustainability of the renewable resource. The basic background of the fishery studied, the most relevant aspects of the methodology used, the main results and conclusions of the study are presented.

Keywords: fishery management; renewable resources; nonlinear programming; total allowable catch quota; planning * Received April 2013. Accepted September 2013

Introduction The lack of ownership of any particular resource is one of the main aspects of the fishing activity. This usually leads to the overexploitation of fishing resources and to overinvestment in fishing effort (overcapacity of the fleet) which are contrary to an efficient development of that activity. When it comes to ensuring sustainability of an activity in the long term, the State needs to participate in fishery resource management, through different control mechanisms that contribute to the biological conservation of marine resources and ecosystems, as well as to the reduction of the economic deficiencies mentioned above. The Operations Research is one of the most reliable methodologies for decision making. It is applied in a wide range of problems arising in different areas and their fields of application also involve the management of natural resources, see e.g. Bjørndal et al. (2012) and Weintraub et al. (2007). In particular, its use in fishery management is adequate to guarantee the sustainability of extractive fishing activities by establishing restrictions that may be easily incorporated in an optimization model, and also to search a pertinent solution by considering criteria that can be defined naturally as an objective function of it, see Bjørndal et al. (2004) and Clark (2005), among others. Therefore, there are numerous references that consider models and strategies for managing a fishery that consider biological population growth models (based on differential equations and econometric models), inserted in optimization models (linear, nonlinear, multiobjective, dynamic, stochastic), optimum control models, computer simulation * Correspondence: Víctor M. Albornoz, Departamento de Industrias. Universidad Técnica Federico Santa María, Av. Santa María 6400. Santiago, Chile. E-mail: [email protected]

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and decision theory models, see Eggert (1998), Batstone and Sharp (1999), Quinn and Derisso (1999), Dew (2001), Leung et al. (2001), Ussif et al. (2003), Escapa and Prellezo (2003), Katsukawa (2004), Ussif and Sumaila (2005), Albornoz and Canales (2002, 2006, 2008), Albornoz et al. (2006, 2009), Hashim (2008), Azadivar et al. (2009), Duncan et al. (2011), Da Rocha et al. (2012), Chakraborty and Kar (2012), Skonhoft et al. (2012) and Waldo and Paulrud (2013), among many others. The knowledge of population dynamics is crucial to represent the future state of a fishery population with respect to a current state. A basic model is the one called a logistic model, which expresses in a simple manner that the biomass level at the end of a period is a function of the biomass at the beginning of the period, the rate of population growth, the maximum biomass level and the capture that has taken place in that period. Extensions of that model follow the same principle but use more complex relations, such as the case of the model structured by age used in this paper, in which the productivity is determined by some stock/recruitments relationship. Nevertheless, the characteristics of the legislation of a fishery for each country, as well as the conditions adopted to represent the environment in which they are inserted, lead to the study and modeling of special considerations. This paper, in particular, presents a nonlinear optimization model as a suitable methodology for determining an optimal annual total allowable catch quota for the Chilean jack mackerel (Trachurus murphyi) fishery, one of the main fisheries in the country. In Chile, the most important instrument of fishery regulation is the establishment of global and individual catch quotas for each species, whose levels are based on a total allowable catch quota and set annually by the Undersecretariat for Fisheries and Aquaculture, in agreement with the legal framework established. Essentially, that regulation establishes an individual quota system that is the result of the annual decision of the total number of tons of authorized capture and certain maximum limits for each company, according to preset criteria and historical background of the operation. Landings of the Chilean fishing sector are important because of the productivity of the coastal marine ecosystem and within the Exclusive Economic Zone of 200 miles. According to the total world landings, in 2010, Chile occupied the ninth place in the world with about 3% of the total and with catches around 6.6 million tons, which represent about 11% of the national annual exports and 3% of the gross domestic product (GDP). Finally, it should be noted that even though the regulations have faced difficulties, it can be seen that the handling of fishery resources has favored improved management and greater innovation by the companies, which have been able to not only plan the size and operation of each fleet better, but also to plan production adequately and incorporate new products with greater added value. The paper is organized as follows: Section 2 briefly describes the characteristics of the species and the basic background of the fishery studied. Section 3 presents a nonlinear optimization model to obtain an optimal annual catch quota, inserted in the regulatory framework above mentioned. Section 4 describes the main results achieved by using the adopted methodology. Finally, Section 5 lists the main conclusions and extensions of this work.

Background of the problem Jack mackerel is a straddling pelagic species whose geographic distribution covers mainly the Southeastern Pacific Ocean (from the Galapagos Islands in Ecuador to the southernmost region of Chile). Two stock units (or selfsustaining biological units) stand out, one of which is distributed along the coast of Ecuador-Peru and the other off the coast of Chile, inside and outside the Exclusive Economic Zone (EEZ). This assumption is based on Serra (1991), SPFRMO (2008) and Ashford et. al (2011) who mention that there is no scientific evidence of a clear interaction between both stock units. Therefore, this fishery is modeled as an independent stock unit. The Chilean jack mackerel fishery is one of the most important resources of the country and takes place in three main zones: northern zone fishery (from the border with Peru to Antofagasta (24°S)), Caldera-Coquimbo fishery (25°- 30°S) and Talcahuano fishery (34°-40°S). In this study, we will denominate north zone the first two fisheries and central-south zone the Talcahuano fishery. In the latter, the jack mackerel fishery has a distribution beyond the EEZ, reaching even 100° west longitude, and includes the international fishing fleet that operates outside the EEZ, see Figure 1.

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Fig. 1. Jack mackerel distribution (Source: Serra and Canales (2010))

In the case of the stock unit in which we are interested, the life cycle of the species begins in the ocean sector of the central-south zone, where it remains during its first year. Then, the young individuals (recruits) start their migratory process from the west to the east, going into high productivity waters in both fishing zones, but concentrating mostly in the northern zone. Here they are exploited by an industrial fleet until they reach the first years of life, when they become sexually mature and begin migrating gradually to the central-south zone, where the adult individuals

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finally are concentrated. These in turn migrate every year in August to the open sea for their reproduction period. Thus, the decisions for catch proposed by the model consider both zones separately, the same as the equations that describes the dynamic behavior of the species, including its migratory behavior. Historically, the central-south zone fishery has concentrated around 90% of the total landings and as well as most of the spawning fish. In fact, in the northern zone, the most widely represented age group in the catches is that of three year old; older adult individuals become less available for the fleet in that zone. Figure 2 shows the proportion of catch-at-age of jack mackerel in both zones.

0.18 C entral-s outh

P roportion

0.15

North

0.12 0.09 0.06 0.03 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ag e (yr)

Fig. 2. Age group structure of Chilean jack mackerel catches (Source: Serra and Canales (2010))

Landings of this resource showed an increasing trend since the beginning of the fishery in the 1970s and until 1995. Then, the catches started to decrease, due to the implementation of fishing regulations and to the foreseen situation of overexploitation of the resource. In 2001, a new Chilean Fishing Law came into force, setting global and individual catch quotas and restricting the access of new operators, thus regulating industrial fishing activity by means of maximum catch limits per shipowner. This administrative measure consists on distributing every year the total allowable annual catch quota allocated to the industrial sector in each zone among the shipowners who have vessels with valid authorizations to conduct fishing activities in the zone, on the basis of previously defined percentages. Prior to this regulation, there was a period in which only a fixed annual catch quota was established and each company competed to get the largest proportion of that quota, this led to an oversize of the fishing fleet situation that the law attempted to correct. The model proposed in this paper must not only take into account the basic characteristics of the population and spatial behavior of the resource, but also ensure that the sexually-mature part of the population (spawning adult biomass) remains above a given level for the sustainability of the activity over time.

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Methodology The total allowable catch (TAC) quota for jack mackerel has been historically proposed on the basis of a stock assessment and a biological reference fishing mortality. That is to say it simply results from selecting a fishing mortality rate that guarantee a given projected biomass value for the current year. On the contrary, the proposed model considers a medium-term planning horizon and provides optimal annual capture quotas for the north and center-south zones that maximize the sum of the captures over the whole planning horizon, considering a factor to give time value to each catch. The decisions satisfy different constraints that express certain limits on the catches for each zone, escape conditions for the survival of the species and numerous population dynamics equations. In particular, the population dynamic model and its initial conditions were taken from Serra and Canales (2010) and correspond to an age structured model and self-generating by means of recruitment of new individuals, depending on the parental biomass (two years out of phase), including also the interaction of the two fishing zones. With the purpose of introducing the optimization model for calculating an annual global catch quota for the Chilean jack mackerel fishery, the components that define the model in terms of the different parameters and decisions considered use the following notation: Parameters

T ρt tr tm wi oi rinorth risouth M Ni,1

:total number of periods (years) of the planning horizon, :a weight that models the value of the harvest at time period t, for t=1,…,T, :the age at which the recruits are incorporated at the beginning of each period, :the oldest age that the resource can reach, :weight of each individual of age group i, for i= tr,…,tm, :the proportion of individuals with sexual maturity at age i, for i= tr,…,tm, :the exploitation pattern from fishing over each age group i in the north zone, : the exploitation pattern from fishing over each age group i in the center-south zone, :natural instantaneous mortality rate, :proportion of individuals of age i, relative to the number of individuals of age t r at the beginning of the planning horizon, :the number of recruits per spawning individual for low biomass levels, :the rate of decrease of recruits when the spawning biomass increases, :proportion of the north zone in the total captures for each year, :factor that defines the part of the year in which spawning occurs and :critical minimum size for spawning biomass.

α β δ γ B

Decisions variables

Ctnorth Ctsouth Nit Ftnorth Ftsouth Zit Sdt

:the weight captured in period t in the north, for t=1,…,T, :the weight captured in period t in the center-south zone, for t=1,…,T, :the total population of age group i at the beginning of period t, for i= t r,…,tm,; t=1,…,T :the fishing mortality rates during period t in the north zone, for t=1,…,T, :the fishing mortality rates during period t in the center-south zone, for t=1,…,T, :the total mortality rates of age group i at period t, for i= t r,…,tm and t=1,…,T, :the total spawning biomass at the end of period t, for t=1,…,T,

The formulation of the proposed model is as follows: T

max

 t 1

 t (Ctnorth  Ctsouth)

(1)

s.t.

Ni ,t  Ni1,t 1e Zi ,t  M  ri

 Zi 1,t 1

north

Ft

north

i  tr  1,..., tm ; t  2,..., T  ri

south

Ft

south

i  tr ,..., tm ; t  1,..., T

(2) (3)

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Ntr ,t  Sdt tr e

 ( Sdt tr )

t  1,..., T

(4)

t  1,..., T

(5)

ri northFt north Z  wi N i ,t (1  e i ,t ) Z i ,t

t  1,..., T

(6)

ri southFt south Z  wi N i ,t (1  e i ,t ) Z i ,t

t  1,..., T

(7) (8)

Sdt  B

t  1,..., T t  1,..., T

Ftnorth ≥ 0, Ftsouth ≥ 0

t  1,..., T

(10)

tm

Sd t   oi wi N i ,t e

Zi ,t

i tr

tm

Ctnorth   i tr tm

Ctsouth   i tr

Ctnorth = δ (Ctnorth + Ctsouth )

(9)

The objective function in (1) corresponds to the maximization of the sum of harvest (in weight) of the north and center-south zones over the whole of the planning horizon, considering a sequence of given decreasing weight {ρt}t=1,…,T that models the value of the harvest over time, assuming that the current harvest is more important with respect to future harvest. On the other hand, the problem involves different constraints that allow representing the population dynamics as well as conditions related to the exploitation of the resource in time. To describe the behavior and inter-temporal development of the resource, a model structured by ages is used, see Haddon (2011). In (2), it is assumed that the number of individuals of a cohort declines exponentially according to the resulting rate of mortality. This rate of mortality is defined in (3) by the sum of the natural instantaneous mortality rate M and the mortality rate from fishing to be determined during each period, considering the exploitation pattern that distributes the mortality from fishing over each age group in the respective fishing zones. In fact, parameters rinorth and risouth represent the relative action of fishery over the structure of ages of the population and migration process of the population (see Fig. 1 and 2). To determine the number of recruits that are incorporated at the beginning of each period t, that is the number of individuals of age i=tr, constraint (4) is based on Ricker's equation (Ricker, 1954), which relates the number of recruits with the spawning adult population in the corresponding gestation period. Thus, (5) defines the spawning biomass registered at a given period as the sum of some fraction of the biomass of adults of each age group that survive one month after spawning. Ending the population dynamics equations, (6) and (7) correspond to Baranov's capture equations (Ricker, 1975), that show the connection between the volume of catches, fishing mortality rates and abundance. From (2), Z the expression N i ,t (1  e i ,t ) represents the number of individuals of age i that die from natural mortality and fishing th during the t period, that multiplied by the fraction of death from fishing at the corresponding zone and the (estimated) weight of each individual gives the catch in weight in that period and zone for age i, that added over the different ages represents the volume of captures each year. Next, the model considers a constraint in (8) that sets a given percentage δ of participation of the north zone in the total capture for each year based on Chilean agreements. Restriction (9) sets the conservation of the resource by means of a spawning biomass ratio at the end of each period greater than or equal to a certain critical minimum size. Finally, in (10) the non-negativity constraints for the fishing mortality rates decision variables are imposed. Summarizing, the model proposed for the planning and control of the Chilean jack mackerel fishery corresponds to a non-convex non-linear optimization model, which proposes an optimum capture solution that attempts to be as high as possible, but guaranteeing at all times the levels required for the conservation of the resource. On the other hand, since the model considers a finite number of periods (years), it must be used on a Rolling Horizon scheme, which is commonly used in inter-temporal planning problems. The model must be solved at the beginning of each year, making the capture decisions for each zone in the first year of the corresponding planning horizon. In this scheme, after one year, a new planning period must be added at the end, and continue making decisions at the beginning of each year that obviously take into account their impact on the future decisions and biomass levels of the species regulated during a new T periods planning horizon.

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Results and analysis The different parameters were provided by the Fisheries Development Institute (IFOP), an agency whose tasks include to provide technical support to the decision-making process, with respect to the annual total allowable catch quota for the Chilean jack mackerel fisheries, as well as in the other fisheries regulated by the State. The dynamic population parameters –relative abundance by age groups, exploitation pattern by zone, parameters of stock-recruitment relationship, fishing mortality by zone, natural mortality, mean weight at age and previous spawning biomass– were obtained from Serra and Canales (2010). Those inputs are results of a stock assessment, approach which involves a statistical model based on the species’ biology and available information of the fishery. This procedure corresponds to the main current used to fisheries management, see Punt and Hilborn (1997), Hilborn (2003) and Mangel et al. (2013), among others. Particularly, the solution and study of model (1)-(10) in this work considered an estimated natural mortality rate M=0.23, a number of recruits per spawning individual α= 6.36238, a rate β=1.20E-07 in the Ricker's equation (4), a factor =9/12 that represents the period of the year in which spawning occurs and a critical population size B=0.4 as proportion of virginal spawning biomass. By assuming a value of tr=2 years for the age at which the recruits reach the fishing zones and tm=12 years as the limit age at which the resource is representative in the landings, the model also considers a relative population size (Ni,1), exploitation patterns for each age and fishing zone (rinorth and risouth), sexual maturity proportion (oi) and estimated weights of each individual of a given cohort (wi) values according to the following table: Table 1. Data of Chilean jack mackerel fishery by age distribution.

Age (yrs)

Ni,1

rinorth

risouth

oi

wi (kg)

2 3 4 5 6 7 8 9 10 11 12

1.00 0.58 0.33 0.19 0.18 0.11 0.11 0.06 0.02 0.01 0.00

0.28 0.81 0.96 0.48 0.10 0.01 0.00 0.00 0.00 0.00 0.00

0.03 0.32 0.89 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.00 0.04 0.50 0.96 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.07 0.13 0.20 0.26 0.33 0.45 0.64 0.88 1.10 1.32 1.65

The rest of the parameters consider a planning horizon of T=10 years, divided into annual periods, a weight in the objective function ρt =1/(1+ρ)t-1 using ρ=0.15 and a participation of the north zone in the total capture δ=10% imposed by Chilean agreements according to historical records. On the other hand, model (1)-(10) has a total of (7+2tm-2tr)T decision variables, (6+2tm-2tr)T-(tm-tr+1) equality constraints and T bound inequality constraints, without considering the non-negativity conditions. In this case, this gives a total of 270 decision variables, 249 equations and 10 inequalities. The model was formulated using the algebraic modeling language software AMPL (Fourer et al., 2003), that contribute to manage and solve the resulting model and the different instances considered as part of the study. The results in this section were obtained with MINOS 5.5 (Murtagh and Saunders, 1998) as a general non-linear optimization solver, which is easily called from AMPL. MINOS applies a project lagrangian algorithm for nonlinearly constrained optimization models, but in view of the non-convex nature of model (1)-(10) there is no guarantee that the algorithm will converge to the global optimal solution from an arbitrary starting point. For that reason, the values shown in this section correspond to the best local minimum found by MINOS given several initial starting points.

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Table 2. Relative catches reached by the model

Period

1

2

3

4

5

6

7

8

9

10

Relative catch

1.00

0.92

1.21

1.18

1.11

1.12

1.26

1.20

1.23

1.31

The results in Table 2 show that the solution reached with the purpose of registering the largest captures as soon as possible has an increasing trend from the first period, but always respecting the condition of a spawning biomass greater than the given critical level. In this context, despite the model suggests initial quotas that are lower than those taken after the third period, the decision allows to increase the quotas in future periods around 30% at the end of the planning horizon. Similarly, it was found that the age groups having the greatest incidence in terms of spawning biomass are those of 4 and 5 years, when 96% of the individuals are sexually mature, so the strategy maximizes the extraction of biomass from this group which has a greater production value. As part of the study, a sensitivity analysis was carried out to analyze the impact produced by values different from those adopted initially in model (1)-(10) for some of the parameters of the problem. More specifically, changes in all parameters related to the management of the fishery and to the natural mortality rate were analyzed. Firstly, the impact on the result for different values of a discount rate ρ was analyzed. This parameter defines the weight ρt = 1/(1+ρ)t-1 in the objective function, which make it possible to understand the differences caused by giving a greater relative value to the catches at the beginning of the planning horizon, compared to those occurring towards the end. The value assumed was ρ=0.15, that after ten periods defines a factor that reduces in 70% the importance of catches. Figure 3 summarizes the changes in the objective function when we increase this value up to 0.40 that, after ten periods, defines a factor that reduces in a 95% the importance of catches, and when it was reduced to 0.0, giving the same relative importance to the period in which the captures are made. In the former, the total capture level during the whole planning horizon is similar to the initial given value and in the later is only about 3% greater than the initial one, but with a different pattern in the catches. In sum, the results show that the objective function value is quite sensitive to the choice of the discount rate ρ, however it has no major impact on the total catch volume over the whole planning horizon.

Fig. 3. Sensitivity on the objective function for different values of ρ

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A second aspect of interest to be studied considered the impact of a higher value for parameter δ, above its present value of 10% share of the global catch quota, fixed by regulation to the industrial fishing sector of the north zone. More precisely, the effect of an increase of parameter δ from its initial value to 12%, 14% and 16%, respectively, was analyzed. The results are summarized in Table 3, which shows that as the share in the captures in the north zone increases, both the value of the objective function and the total captures in the whole planning horizon decrease. This reduction is associated with the increasing fishing mortality, particularly in the pre-spawning individuals. In view of the situation observed, the effect of relaxing that 10% share participation in the north zone in each period was also studied. As a result of the substitution of equation (8) in the model by an inequality that provides a quota of at most 10% for the north zone in each period, it was possible to obtain a solution that exceeds the total captures during the planning period with respect to the solution found initially, with an increase in the value of the objective function somewhat greater than 1%. Furthermore, simply eliminating constraint (8), a solution was found, which significantly exceeds the total of the captures obtained with respect to the initial formulation, where the increase in the value of the objective function is close to 3.6%. All of the above does not advise an increase in the share of the north zone and shows that this fixed percentage is clearly suboptimal. Table 3.Variation of share in the north zone

δ =10% δ =12% δ =14% δ =16%

Objective function value

Relative catches

9.379 9.361 9.343 9.324

1.000 0.997 0.993 0.990

Another aspect of interest in the sensitivity analysis considered a set of scenarios with a higher value for the minimum level of spawning biomass, which appears in constraint (9), to guarantee the conservation of the resource, whose initial imposed value was pertinent but maybe somewhat lower than one that generates the greatest production value during the whole planning horizon. Table 4 shows that when we increase the value of B up to 0.46 (an increase of 15%) all the solutions obtained provide a worse objective function value and lower total catches. However, in relative terms the total catches during the whole planning horizon are quite less sensitive, which indicates that – also from a production point of view– a greater value of B is a good precautionary measure for its conservation. Table 4. Variation in the minimum level of spawning biomass B

Objective function value

Relative catches

0.40

9.379

1.000

0.41

9.312

1.000

0.42

9.242

1.000

0.43

9.171

0.999

0.44

9.095

0.998

0.45

9.009

0.995

0.46

8.921

0.992

Finally, alternative scenarios were studied to reflect the sensitivity of the model to variations in the value of the most important parameter in the dynamic population equations: the natural mortality rate M, often assumed as a constant based on some biological estimation. This parameter is very important because somehow it is present at every constraint in the model and defines the scale of population productivity. Table 5 summarizes the objective value function and total catches when we consider different values close to the adopted one (M = 0,23). The results show that the behavior is quite sensitive to the given value, with relative variations similar to those introduced in

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this parameter and shows the convenience for taking into account this variations in some extensions, where the proposed model can serve as a base for the formulation of more complex models that, however, is not within the scope of this research work. Table 5. Variation in the minimum level of natural mortality

M

Objective function value

Relative catches

0.20 0.21 0.22 0.23 0.24 0.25 0.26

10.66 10.33 9.90 9.38 8.86 8.34 7.83

1.11 1.09 1.05 1.00 0.94 0.89 0.84

Conclusions The results of this work have made it possible to establish that the use of a non-linear optimization model, combined with the use of a population dynamics model with an age structure and self-generating by means of recruiting, provides a useful tool for the analysis, control and sustainable as well as efficient exploitation of the fishery resource studied. The analysis carried out makes it possible to reach conclusions about various aspects that contribute to define or reinforce some policies for the management of the resource: first, it is found that increasing the share of the north zone (equivalent to increasing the exploitation of juveniles), leads to less efficient solutions and to a systematic reduction in the optimum value. Furthermore, relaxation or elimination of that share leads to better solutions than those found for the initial model, allowing the conclusion to be reached that at least when faced with any proposal for increase in the exploitation of the resource should not be approved. Next, it should be noted how pertinent it would be to increase the given minimum spawning biomass level because when increasing that level, it is possible to define a better precautionary measure that cause a non-significant reduction in the volume of total catches. On the other hand, it should be drawn to attention how sensitive the behavior of the optimal value is to variations in natural mortality –the key parameter of the dynamic population- , so choosing this parameter is highly relevant in the solution of the model. Further to the results and satisfactory performance shown by the proposed model, it should be mentioned that as a future development for its improvement, the explicit inclusion of probable scenarios for this key parameter is quite desirable for errors in the estimation and/or uncertainty in the future behavior of the marine ecosystem, that would give rise, for example, to a stochastic optimization model that is not within the scope of this work. Acknowledgements. The authors would like to thank the anonymous referees for their valuable suggestions on an earlier version of the paper and the Editors of the special issue: OR Applications in Natural Resources Management. Financial support from DGIP (Grants USM 28.13.69) and CIDIEN of the Universidad Técnica Federico Santa María is gratefully acknowledged.

References Albornoz VM and Canales CM (2002). Planificación de la Conservación y Explotación del Langostino Colorado usando un Modelo de Optimización Estocástica No - Lineal con Recurso. Información Tecnológica 13: 183-190. Albornoz VM and Canales CM (2006). Total allowable catch for managing squat lobster fishery using stochastic nonlinear programming. Computers & Operations Research 33: 2113-2124.

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