Optimal Harvesting Strategies for a Multi-cycle and Multi-pond Shrimp Operation - A Practical Network Model

Mathematics and Computers in Simulation 68 (2005) 339–354

Optimal harvesting strategies for a multi-cycle and multi-pondshrimp operation: A practical network model

Run Yua,b, PingSun Leungb,∗

a Department of Economics, University of Hawaii at Manoa, Honolulu, HI 96822, USAb Department of Molecular Biosciences and Bioengineering, University of Hawaii at Manoa,

3050 Maile Way, Gilmore 111, Honolulu, HI 96822, USA

Received 18 June 2004; received in revised form 20 January 2005; accepted 31 January 2005Available online 7 April 2005

Abstract

In this paper, we introduce a network formulation of the optimal scheduling model for a multi-cycle and multi-pond shrimp operation grounded on the original optimal harvesting theory for a single production unit. The optimalschedule comprises the harvesting and restocking time that maximizes total profit throughout the planning horizon,bounded by the underlying biological and economic conditions. The model takes into account the informationsuch as harvest size distribution, seasonality of price, temperature and weight-dependent growth, and labor forceand market demand constraints. We applied the model to an existing shrimp operation in Hawaii with 40 one-acreponds and generated the optimal schedule for a year that maximizes overall production. The model schedule isfound to be able to increase total production by 5% when compared to the schedule generated using an “educated”trial-and-error procedure currently practiced by this operation. Further insights for this multi-cycle and multi-pondscheduling problem were also generated through several alternate simulations. It is found that labor force and marketdemand constraints are the key factors in scheduling multi-cycle and multi-pond shrimp operations.© 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:Optimal production schedule; Shrimp farming; Aquaculture; Network model

∗ Corresponding author. Tel.: +1 808 9568562; fax: +1 808 9569269.E-mail addresses:[email protected] (R. Yu), [email protected] (P. Leung).

0378-4754/$30.00 © 2005 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2005.01.018

340 R. Yu, P. Leung / Mathematics and Computers in Simulation 68 (2005) 339–354

1. Introduction

The problem of determining the optimal harvest size of farmed fish has first been analyzed and modeledby Bjorndal[2] using an optimal control framework for salmon culture. Arnason[1], Heaps[8,9], andHean[7] have extended this model to include optimal feeding schedule, density-dependent growth andmortality and potential culling, and release cost, respectively. Cacho et al.[3] have applied a similaranalytical framework to identify the feeding strategies of catfish culture. Springborn et al.[18] haveanalyzed the optimum harvest time for cultured Nile tilapia using a similar framework. Leung et al.[14] have also employed a similar framework to identify the optimal harvest age for giant clam culture.Recently, Pascoe et al.[15] have applied a similar framework to identify the optimal harvest time forsea bream and tiger prawn. Forsberg[4,5] has proposed an alternative multi-period linear programmingapproach to model explicitly the size-structure of farmed fish in determining the optimal stocking andharvesting schedules.

In terms of applications to crustaceans, Leung and Shang[12] have applied Markov process anddynamic programming to model the stocking and harvesting decisions for freshwater prawns. Karp etal. [11] have also applied dynamic programming to determine the optimal stocking and harvesting ratesof shrimp in the Southwest U.S. Hochman et al.[10] and Leung et al.[13] have extended the stochasticgrowing inventory framework to marine shrimp culture for the purpose of identifying the optimal stockingand harvesting schedules. Tian et al.[20] have evaluated the implications of using different shrimp growthfunctions on optimal harvest size. Purwanto[16] has developed a set of linear and non-linear programmingmodels for optimizing the economic benefits from shrimp farming in Australia and West Java, Indonesia.Tian et al.[19] have also developed a computer simulation model to analyze the economics of shrimpproduction under different stocking regimes, harvesting schedules and farm sizes.

While the previous research mentioned above has provided the theoretical foundation to model optimalharvesting decisions for finfish and shrimp culture, they are generally not readily applicable to commercialoperations for the following reasons. First, most of the models described above used experimental data tomodel the growth process and they have been shown to differ widely from actual commercial operations.For example, the model by Leung et al.[13] was based on limited data from one growout cycle of marineshrimp in an experimental pond. Second, all of the models are designed primarily as a research inquirytool rather than an operations management tool for on-farm applications. This is particularly true for themodel by Leung et al.[13] where it has been used primarily to investigate the economics of controlledenvironment. Third, almost all of the models were developed for optimizing the harvest schedule of asingle pond or production unit. While it is certainly important to derive and understand the essence ofthe optimal harvest schedule of a single pond, it is essential to extend this framework into a whole farmsetting whereby all the ponds or production units are scheduled in an optimal manner to smooth outproduction so as to satisfy the labor force and market demand constraints. Against this background, thepurpose of this research is to develop, test and operationalize a management model that takes into accountthe multi-cycle and multi-pond nature of most commercial shrimp farms in order to solve for the optimalgrowout production schedule of the entire farm.

The paper is organized as follows: Section2 presents the mathematical representation of the optimalscheduling problem and introduces the underlying biological and economic functions. Section3 providesthe network representation of the optimal scheduling model that is equivalent to the otherwise unsolvablemathematical formulation. Section4 applies the network model to an existing shrimp farming entitywith 40 growout ponds to generate the optimal schedule for a year. Section5 discusses the results of

R. Yu, P. Leung / Mathematics and Computers in Simulation 68 (2005) 339–354 341

the optimal scheduling and implications from several alternate simulations. Section6 concludes thepaper.

2. The mathematical formulation

In this section, we present the mathematical formulation of the optimal scheduling problem and explainthe underlying biological and economic functions involved.

Our whole-farm optimal scheduling model is grounded on the optimal harvesting theory pioneered byBjorndal[2]. The problem for the planner at hand is to determine the schedule of harvesting and restockingtime that maximizes total profit for a multi-pond shrimp operation for a finite planning horizon that canaccommodate several production cycles.

In order to provide insight into this problem, we start analyzing our whole-farm optimal schedulingproblem by considering a single representative pond. The finite planning period is continuously num-bered in the time decision unit, e.g. a day or a week, ort= 1, 2, 3,. . ., T. T is the ending time whenall shrimps are assumed to be harvested. DefineT = {t|t = 1, 2, 3, . . . , T }. The schedule for a singlegrowout production cycle can be represented by a vectorh(i, j), i, j ∈T. h(i, j) defines a single growoutproduction cycle that begins at timei and ends at timej. Define the seth = {h(i, j)|i, j ∈T; i < j}. his composed of all feasible schedule for a single production cycle. As a result, the multi-cycle produc-tion schedule for the whole planning period can be constructed as the combination of elements inhthat does not conflict with the time path. LetHl be such a combination that consists ofm elements(h1(i1, j1), h2(i2, j2), . . ., hm(im, jm)) from the seth. Rank these vectors according to their first ele-menti in the bracket from the smallest to the largest, the non-conflict time path condition requires that1 ≤ i1 < j1 < i2 < j2 < · · · < im < jm ≤ T . Following the definition ofh(i, j), i’s andj’s represent theplanting and harvesting time, respectively. The non-conflict time path condition simply ensures that pro-duction cycle cannot overlap each other. Therefore, the feasible multi-cycle schedule set can be definedas follows:

H = {H(h1(i1, j1), h2(i2, j2), . . . , hm(im, jm))|h(i, j)

∈h, 1 ≤ i1 < j1 < i2 < j2 < · · · < im < jm ≤ T, m ∈N+}Each element inH represents a feasible multi-cycle schedule that covers the whole planning period.For example,H(h(1, 100), h(105, 230), h(T − 100, T )) denotes a management schedule that consistsof three production cycles that begin at time 1, 105, andT− 100 and end at time 100, 230, andT,respectively.

Now we proceed to construct the planner’s objective function through introducing the underlyingspecific biological and economic functions.

2.1. The growth function

The fundamental factor in production scheduling is the growth function. As many studies suggested,the incremental shrimp weight can be modeled as a function of various variables such as feeding, stockingsize, biomass, density and temperature[4,15,17,21]. In practice, if we assume that controllable variablessuch as feeding, biomass, and stocking density are already at the optimal levels, the incremental shrimp


weight can be treated as if it only depends on temperature and the previous period’s weight that can beexpressed as follows:

Wi,t+1 = Wi,t + G(Tt, Wi,t) (1)

whereW is the average weight of farmed shrimp measured at the beginning of each discrete time; its firstsubscripti denotes the planting time and the second subscriptt denotes current time;G(Tt, Wi,t) is thegrowth increment during timet to t+ 1. Therefore, expression(1) states that the current average shrimpweight equals the average shrimp weight at the preceding time plus the growth increment. As a result,for a feasible production cycle scheduleh(i, j), the average weight of the harvested shrimp is equal to

Wi,j = Si +j−1∑t=i

G(Tt, Wi,t) (2)

whereSi is the average shrimp weight when they are planted at timei; Si ∈S, or S= {Si | i ∈T}. Theelements inSare the average weight of shrimp available for planting at each discrete time.

2.2. The survival function

Similar to the growth function, the number of surviving shrimp can be expressed as

Ni,j = Ni,iMi,j(Bi,j) (3)

whereNi,j is the number of surviving shrimp at harvesting timej; Ni,i is the number of shrimp at plantingtime i. The survival rateMi,j is a function of biomass,Bi,j and is assumed to decrease with an increase inbiomass.

2.3. Size distribution function and price function

The above growth function actually focuses on the average weight of a shrimp cohort. However, theactual harvested shrimps vary in size. Moreover, larger size of shrimp commands a higher price. As aresult, harvest size distribution has an important impact on optimal scheduling[19]. Furthermore, in manysituations, price is characterized by seasonality[14]. In order to capture these effects, size distributionfunction and price function are introduced, respectively. Distribution functionδk

t (Wi,t) is the percent oftotal harvested shrimp number that falls into thekth size category. It is a function of the average shrimpweight. The parameterk represents the category of shrimp size, e.g. size counts of 21/25, 26/30, 31/35and 36/40. The corresponding price function for thekth size category isPk

t (k, t). Basically, it is a time-dependent function and thus captures the seasonality effect. The price for larger size shrimp is assumedto be always higher during a particular season.

2.4. Cost function

The total production cost is composed of variable costs and fixed costs and is expressed as follows:

Ci,j =j−1∑t=i

Ni,tPf,tEtG(Tt, Wi,t) + OVC(i, j) + FC(i, j) (4)


where the first component on the right hand side is the feed cost function;Pf,t is the unit price for feedsat timet; Et is the food conversion efficiency at timet, which denotes the units of feeds required for oneunit of growth.OVC(i, j) represents variable costs other than feed costs andFC(i, j) is the fixed cost forone production cycle. The feed pricePf can be modeled as a time-dependent function to take into accountseasonality of price. Also, the food conversion efficiencyEt can be modeled as a function of weight andtemperature.

2.5. Market value function and profit function

We now turn to the derivations of the market value for every shrimp size category and the profit for asingle harvest, which can respectively be expressed as Eqs.(5) and (6)as follows:

Vki,j = Pk

j Ni,jδkj(Wi,j), k = 1, 2, . . . , K (5)

Vi,j =K∑

k=1

Vki,j − Ci,j (6)

Finally, total profit for a feasible multi-cycle scheduleHm can be expressed as

NVHm=

∑h(i,j) ∈ Hm

(Vi,j − Ci,j), Hm ∈H (7)

We now proceed to the scheduling of several production ponds simultaneously. The multi-pond schedulingwill not differ from single pond scheduling in nature, if there is no constraint on the frequency ofharvesting and restocking. In that situation, the optimal scheduling can be obtained by applying thesingle pond scheduling model to each pond respectively. However, the multi-pond scheduling problembecomes complicated and different from sequentially solving the single pond scheduling problem whenthe frequency of harvesting and restocking are restricted due to the limitation of labor force and marketdemand.

2.6. The labor force and market demand constraints

In practice, due to the limitation of labor force, facility capacity and market demand, shrimp farmerscannot harvest shrimp unlimitedly at any one time. Furthermore, shrimp farmers would also tend to spreadtheir production throughout the year in order to better utilize the available labor force as well as to smoothout their supply to meet market demand and cash flows. We introduce these constraints into the modelsimply in terms of minimum and maximum frequency of harvesting and restocking. The constraints canbe expressed as follows:

for shrimp planting at timei CS min ≤N∑

n=1

hn(i, j) ≤ CS max (8)

for shrimp harvesting at timej CH min ≤N∑

n=1

hn(i, j) ≤ CH max (9)


wherehn(i, j) denotes a production cycle vector for thenth pond which begins at timei and ends at timej.CSmin, CSmax, CHmin and CHmax are constants depicting the maximum and minimum number of plantingsand harvestings. Expression(8) shows that at any one time, there must have at least CSmin plantings butnot to exceed CSmax plantings. Similarly, expression(9) ensures that at least CHmin harvestings to beperformed at any one time but there can be no more than CHmax harvestings.

2.7. The optimal scheduling problem

As stated before, the objective of the planner is to maximize profit from production of all pondsthroughout the entire planning period. Therefore, the multi-cycle and multi-pond optimal schedulingproblem can simply be viewed as choosing one element from each pond’s feasible multi-cycle schedulingsetHn such that their combinations can result in maximum profit under conditions restricted by expressions(8) and(9). The overall maximization problem can now be expressed as

MaxN∑

n=1

∑H ∈ Hn

(K∑

k=1

Vkn,i,j − Cn,i,j

)

s.t.

CS min ≤N∑

n=1

hn(i, j) ≤ CS max for i ∈T

CH min ≤N∑

n=1

hn(i, j) ≤ CH max for j ∈T

(10)

In order for the present model to be of practical use, we will have to transform the mathematicalmodel into an available commercial solver in order to solve the optimal scheduling problem. However,we will encounter two major difficulties inherent in this mathematical formulation. One difficulty is theintrinsic nonlinearity of the various underlying functions such as the growth function and the harvestsize distribution function. Even if we can overcome this nonlinearity issue by either linearizing thenonlinear functions or simply adopting a nonlinear programming framework, we will still be unableto solve this inherently difficult 0–1 integer programming formulation. Suppose we let a 0–1 binaryvariable to represent the management decision where “1” denotes harvesting/restocking to occur and “0”denotes otherwise. Using that representation, the feasible scheduling set will consist of 2NT elements fora scheduling problem forN ponds andT time units. For example, even solving a 1-year weekly planningof 10 ponds will be prohibitive in terms of computing time even using the most efficient branch andbound technique and on the fastest computer. This well-known NP hard problem of integer programminghas rendered such formulation to be impractical. On the other hand, the network-related construction hasbeen well documented to be an efficient approach to handle this kind of integer discrete problem[6].Therefore, we will turn to the network representation of our optimal scheduling model for a practicalsolution strategy.

3. The network representation

As we have done previously, we will introduce how to convert our mathematical formulation into anetwork representation by first considering a representative pond.


Fig. 1. Network structure of a representative pond.

Fig. 1 illustrates the network structure for a representative pond. The node (circle) represents onediscrete time in the planning horizon when a managerial decision can take place. Node 0 denotes thetime when the planner prepares the schedule. The series of nodes from 1 toT represents the setT. Thearc from each node to each later node represents a potential single production cycle schedule where thestarting node denotes the planting time and the ending node denotes the harvesting time. As a result, eacharc corresponds to an element in the seth. Furthermore, a feasible multi-cycle management schedule isthe connection of arcs that does not overlap each other. For example, inFig. 1, the connection of arcs a,d, g and j and the connection of arcs a, e and i are feasible multi-cycle management schedules, while theconnection of arcs b, d, g and j and the connection of arcs a, e, g and j are not feasible, because arc boverlaps arc d and arc e overlaps arc g.

Fig. 2. Representation of network structure in a decision tree form.


Fig. 3. An example of a feasible multi-cycle schedule. Node conservation requirement: the flow on every arc must be either 1or 0. Flow conservation requirement: sum of inflows equals sum of outflows for every node.Note: the flow is counted as inflowfor ending node, while counted as outflow for beginning node. Hence, the status of inflow represents the harvesting decision andthe status of outflow represents the restocking decision.

Fig. 2 illustrates the network’s managerial process in the form of a decision tree where theith nodehas (T− i) branches. At each node, we will choose one and only one branch.NV(i, j)s are the resultingprofit related to the corresponding decision branches. In order to maximize total profit, we will choosethose branches so that the sum of their profit will be the greatest.

In this network representation, the connection of arcs that makes a feasible multi-cycle schedule isidentified by two requirements: the flow conservation requirement and the node conservation requirement.Flow is defined as the decision status of an arc. In our model, there are two possible statuses for each arc.We define “1” to be the status of “choosing as part of schedule” and “0” to be the status of “discarding aspart of schedule”. As a result, the flow conservation requirement requires that the flow on every arc mustbe either 1 or zero. The node conservation requirement requires that the sum of flows into every node(inflows) must equal the sum of flows out from that node (outflows). Consequently, in our problem thenet flows (inflow minus outflow) of node 1 to nodeT− 1 are all zero and there is one outflow emergingfrom node 0 and one inflow into nodeT. A flow path that starts from node 0 and ends at nodeT andabides the node conservation requirement and flow conservation requirement defines a potential multi-cycle management schedule. InFig. 3, the number above the arc denotes the status of flows. Underthe two conservation requirements, every node at most has one inflow and one outflow. Once a nodehas an inflow, it must also have an outflow. The feasible multi-cycle schedule presented inFig. 3 isH((0, i), (i, T − k), (T − k, T )). The optimal scheduling problem for a single pond is in fact equivalentto determining the flow path that maximizes total profit for that pond.

Fig. 4 illustrates how we can compute the resulting profit for a potential harvesting through theunderlying biological and economic functions implemented using a set of spreadsheet LOOKUP ta-bles. For each arc, there is a corresponding profit (or loss). Therefore, by multiplying every arc’s flowwith its profit, we can obtain the profit for a flow path that defines a feasible multi-cycle managementschedule.

We now consider managingN ponds together and thus we need to incorporate the labor force andmarket demand constraints. The labor force and market demand constraints are introduced into the modelthrough the node-flow conservation requirement. In the network representation, the arc is counted as an“outflow” to its beginning node, while it is counted as an “inflow” to its ending node. Since the beginningnode represents the restocking time and the ending node represents the harvesting time, an outflow thusindicates a managerial decision on restocking and an inflow indicates a managerial decision on harvesting.In consequence, the labor force and market demand constraints or the restrictions on harvesting andrestocking can be imposed upon the sum of inflows and outflows of each node, respectively. Note that anode can be a beginning node and ending node at the same time. For example, inFig. 1, node 2 is the


Fig. 4. Evaluation of profit through the underlying biological and economic functions within the scheduling problem.

beginning node for arc f, while is the ending node for arc b. Hence, for node 2, arc b is its inflow, whilearc f is its outflow. For our present situation, node-flow conservation requirement specifies that the sum ofinflows of every node except node 0 across all ponds must abide the minimum and maximum harvestingfrequency constraints and the sum of outflows of every node except nodeT across all ponds must abidethe minimum and maximum restocking frequency constraints. It is rather straightforward to obtain boththe sum of overall inflows and the sum of overall outflows for every node by directly adding the inflow andoutflow across all ponds respectively and implement the constraints. The solution of optimal schedulingwith the labor and market demand constraints is obtained by finding the feasible combination of flowpaths for all ponds that produces the highest profit.

4. An application

In this section, we apply our optimal scheduling network model to an existing shrimp farming entity totest its applicability, validity and reliability. This commercial shrimp farm under investigation currentlyoperates 40 one-acre ponds throughout the year. Presently, the year ahead harvesting and restockingschedule is decided by a manager through an “educated” trial-and-error process. So, we will test ourpresent network model based on the same information as used by this manager in order to compare themodel schedule and the schedule developed by this manager.


Fig. 5. The network structure: an application.

Specific assumptions, background information and functional forms for this application are asfollows:1

(1) All of the 40 production ponds are homogeneous in terms of environmental and production conditions.The production schedule comprises 52 weekly managerial decisions for a year. Harvesting is assumedto occur only on Mondays, Wednesdays and Thursdays while planting to occur only on Fridays aspresently done on this farm. To capture this characteristic, we use subscript a to denote the potentialharvesting days (Mondays, Wednesdays and Thursdays) and subscript b to denote the potentialrestocking day (Fridays) for each week. We define these two types of node as harvesting node andrestocking node respectively. As a result, two types of arcs are allowed in this present network (seeFig. 5). One is the preparation arc, which starts from a harvesting node to another later restockingnode and defines the resting time before a new planting occurs. For example, arcs (11a, 14b) and (31a,33b) in Fig. 5 are preparation arcs. The other is the production arc, which starts from a restockingnode to another later harvesting node and defines a production cycle. For example, arcs (14b, 31a) and(33b, 51a) in Fig. 5are production arcs. The resting time between two production cycles is restrictedto range from 1 to 4 weeks, according to this farm’s present practice. As a result, the preparation arcconnects each harvesting node to each later restocking node that is at least 1 week ahead but no morethan 4 weeks ahead. For example, the arc (11a, 14b) in Fig. 5 implies that the farm keeps the pondempty for 3 weeks before a new production cycle starts. As for the arc that starts at node 0, it willend at a node with subscript b if the pond is empty at initial planning time. Otherwise, it will end ata node with subscript a. Node 53 denotes the ending time. All ponds are assumed to be harvested atthat time. Hence, all flow paths will end at node 53.

(2) The growth function is assumed to depend only on weight and temperature, because the amount offeeding is assumed to be at optimal levels in order to maximize growth given weight and temperature.This shrimp farm has already generated a weekly growth chart based on its historical sampling data.This chart provides the growth information in this application. This growth function has two essentialfeatures that significantly influence our optimal managerial schedule. One feature is the seasonalityof growth caused by seasonal temperature differences. Incremental growth generally increases fromspring to summer, reaches its peak in May, and then decreases gradually until it reaches its trough inDecember. The other feature is related to the weight dependency in that incremental growth declinesonce the shrimp grows up to 21 g.

(3) Harvest is assumed to occur when the average weight of shrimp falls between 21 and 25 g. Thisreflects the current practice of the market targeted by this operation. This feature is incorporated byassigning zero to production when the weight deviates from the range of 21 to 25 g.

(4) Average shrimp weight for planting is assumed to be identical and equals to 2 g.

1 In order not to reveal any confidential information of this commercial shrimp farm, we will be presenting many of theinformation in a generic form just for the purpose of illustration.


(5) For our present scheduling application, we used a recent historical snapshot from this farm’s recordas the starting point. Hence, the 40 ponds are assumed to be at different points during their productioncycles at the initial planning time. In particular, 36 ponds have already been stocked with shrimpsand 4 ponds are empty.

(6) Average stocking number is assumed to be identical and equals 400,000 per pond. Survival is assumedto be 50% from stocking to harvest size between 21 and 25 g.

(7) Prices and costs were both ignored in this present application. Thus we are in essence only maximizingtotal production and not profit in this application. This simplified assumption is implemented merelyfor illustrative purpose in testing this model so as not to unnecessarily disclose confidential informationof this farm.

(8) No more than three harvests per week can be performed and there must have at least one harvest perweek. There is no specific restriction on restocking.

It should be noted while some simplifying assumptions were made in this application for illustratingthe working of this model; it nevertheless reflects very closely the current practice of this farm accordingto its vice president.

5. Baseline and alternative simulation results

The above network model consists of 30,000 binary variables and is implemented on MS Excel usingFrontline’s Large-Scale Linear Programming Solver. It takes several hours for Solver to crunch the optimalmanagerial schedule.

Fig. 6 provides the optimal model managerial schedule for the planning horizon of 1 year for thebaseline simulation. All harvests are guaranteed to result with shrimp within the target size range by thisschedule. In contrast, the existing “educated” trail-and-error managerial schedule could not provide sucha guarantee. The total production under the optimal model schedule is estimated to be 5% higher than thatbased on the existing schedule. This estimated increase is equivalent to production from an additionaltwo ponds which is not an insignificant amount.2

In order to extract some insights from the optimal scheduling problem, several other scenarios are alsosimulated, including the scenario where all ponds are empty at the initial planning time and the scenariowhere no labor force and market constraints are applied. Several interesting findings can be discernedfrom comparing the results of these simulations and they are summarized as follows.

(1) Even without considering the labor force and market constraints, the optimal harvest size variesdepending on the assumption of the length of preparation time. Suppose at the end of timet, theaverage shrimp weight firstly reaches the target size range, e.g. 21 g, so that we can harvest at thebeginning of timet+ 1 and start a new cycle after a rest period or keep growing the shrimp to a largersize at timet+ 1. The expected benefits will beNG21+

t+1 if we keep growing the shrimp to a larger size

2 It should be noted that the management of this operation is very experienced and highly sophisticated. Thus the trial-and-errormanagerial schedule is certainly not “crude” but rather a very “educated” schedule. While we could not reveal the exact dollaramount of the 5% difference, it is sizable and the management has also indicated to us that our present model is certainly worththe effort particularly the model would provide a flexible and automatic tool for their periodic scheduling task. This would alsoimply that operations with less sophisticated management skill would even benefit more from the present model.


Fig. 6. The optimal managerial schedule.

while the opportunity cost will beNG21−t+i if we harvest the shrimp at timet+ 1 and start a new cycle

at timet+ i (whereN is the surviving shrimp number and assumed to be identical). If the next cyclecan be implemented without any preparation time, ori = 1. That is the new cycle starts at timet+ 1. Inthis case, the expected benefits will beNG21+

t+1 and the opportunity cost will beNG21−t+1 , respectively.

Because the assumed growth function specifies that smaller shrimps grow faster than larger ones (i.e.,G21+

t < G21−t for all t), thenNG21+

t+1 < NG21−t+1 . Therefore, once the average shrimp weight reaches

the target range (21–25 g), we should harvest the shrimp immediately, because the expected benefitsof harvesting shrimp at a larger size at next time period is always less than the opportunity costs. Sincethe smallest target weight is 21 g and the average weekly incremental weight is roughly 1 g, we canderive that the optimal harvest size will fall between 21 and 22 g if the next cycle can be implementedwithout any preparation time. However, due to seasonal variation in growth rate,G21+

t+1 maybe greaterthanG21−

t+i , if the assumed resting period is between 1 and 4 weeks, or the new production muststart at least 1 week after and within 5 weeks (i.e., 2≤ i ≤ 5) as in our present problem. In this case,the optimal harvest size may fall into the range of 22–25 g. Results of the simulations (seeTable 1)indicate that the dominant optimal harvest size is between 21 and 22 g. In fact, 68% of the all 106harvests fall within this range. The optimal harvest size of the remaining harvests ranges from 22to 25 g.

(2) The optimal harvest sizes of the baseline are found to deviate from those when there are no laborforce and market demand constraints.Fig. 7 shows that when there are no labor force and marketdemand constraints, the optimal harvest schedule is not evenly distributed throughout the year withmany weeks of no harvest as well as many weeks with more than three harvests. Obviously, withthe imposition of the labor force and market demand constraints, some production cycles would


Table 1Comparison of managerial schedule with/without the labor force and market constraints

With constraints Without constraints

Total number of harvests 99 106Average number of weekly harvests 2.1 2.2

Standard deviation of number of weekly harvests 0.96 2.81Weeks with no harvest 0 23Weeks with 1–3 harvests 52 13Weeks with 3+ harvests 0 16Maximum number of weekly harvests 3 15

Average weight at harvest (g) 23.5 23.2Number of harvests with average weight 21–22 g 54 72Number of harvests with average weight 22–23 g 16 11Number of harvests with average weight 23–24 g 10 11Number of harvests with average weight 24–25 g 19 12

Decrease in total production (%) 5

have to be extended and hence with larger harvest sizes than the situation without the constraints.Results of our simulations indicate that overall production would be reduced by an estimated 5%level.

Fig. 7. The optimal managerial schedule without labor and market constraints.


(3) The labor force and market demand constraints dramatically change the overall optimal managementstrategies and they are considered the critical factors of the multi-pond and multi-cycle schedulingproblem. Table 1 provides a comparison of the managerial schedule with and without theseconstraints. Without these constraints, the average number of harvests per week is 2.2 and itsdistribution is greatly uneven. The schedule indicates that there are 23 weeks with no harvest at alland 16 weeks with more than 3 harvests. Especially, in week 36, it has 15 harvests. On the otherhand, with the labor force and market demand constraints, the average number of harvests per weekis reduced to be 2.1, because fewer harvests are allowed within the year. The derived scheduleshows that there are 23 weeks with three harvests and 19 weeks with one harvest. The smoothingof production to meet labor requirement and market demand throughout the year is certainly morerealistic and representative of actual shrimp farm operations. However, as mentioned in (2), thiscomes at a cost of roughly 5% of production reduction.

(4) The model managerial schedule is very susceptible to the initial stocking conditions. We may not beable to arrive at a feasible managerial schedule under certain initial stocking conditions. One obviousexample is the scenario where all ponds are empty initially. It is easy to see that the condition ofat least one harvest per week cannot be satisfied at all. Similarly, because the target shrimp size isbetween 21 and 25 g and most weekly growths are greater than 1 g, given the constraint of at mostthree harvests per week as in our present application, it is very possible to fail to find a feasiblemanagerial schedule when there are many ponds stocked with similar shrimp size initially. Forexample, the scenario where 15 ponds have their shrimp sizes between 5 and 6 g initially fails togenerate a feasible managerial schedule. Our simulations, using the actual operational data, however,do not encounter this problem. It implies that this commercial shrimp farm actually operates thegrowout entity quite efficiently through trial and error and/or learning by doing.

In summary, our model managerial schedule outperforms the farm’s existing schedule in three respects.First of all, our model managerial schedule guarantees that all harvests will result with shrimp within thetarget size range. Second, under the model managerial schedule, the harvest shrimp size is essentiallydetermined by weighting between the expected benefits and opportunity costs while under the existingschedule, there is no explicit consideration of benefit and cost at all. Finally, the model managerial scheduleis generated by systematically maximizing the overall production and thus incorporates the interactionsbetween ponds due to the labor and market constraints, while the existing schedule can not systematicallysynchronize these interactions at all. Hence, it is not surprising that our model managerial schedule canimprove the farm’s performance as our simulations documented.

6. Concluding remarks

In this paper, we introduce a practical scheduling model for a shrimp operation practicing multiplecycles per year in a multiple ponds setting to derive the optimal harvesting and restocking schedule. Byconverting the scheduling problem into a network formulation allows us to solve this otherwise unsolvablehard integer programming problem. The successful application to an existing shrimp operation in Hawaiihas illustrated the model’s ability and reliability in improving the quality of scheduling decisions andthe resulting production for shrimp culture. The network formulation of this scheduling problem can beconcluded to be a useful tool for supporting management decisions. The present model can be applied


to other aquaculture operations utilizing multiple production units without changing much of the basicstructure since the model is constructed in a very general form as described in Sections2 and 3. It shouldbe noted that many of the model features have not been utilized in the present application which maybe deemed essential in applications in other settings such as the seasonality effects of price and demandon the optimal schedule. Further experimentation with the present model will no doubt provide furtherinsights in improving the present model structure.

Besides using the model as an operating tool in deriving the optimal harvesting and restocking schedule,the present model can also be useful as a design tool such as to identify the optimal number of growoutponds given the farm’s labor and market demand constraints. The present model can also be extendedto include the nursery and hatchery phases of shrimp farming so as to synchronize production betweenthese and the growout phase. The present modeling framework certainly provides a solid foundation forthese and other possible model extensions for improving the management of aquaculture operations.

Acknowledgements

The authors thank Dr. Paul Bienfang of CEATECH USA, Inc. who has contributed significant insightsinto the construct of the present model and also for providing valuable farm information without whichthis study would not be possible. This study was made possible by the generous support of the UnitedStates Department of Agriculture Cooperative State Research, Education and Extension Service (USDAgrant 2003-34167-14034).

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Optimal Harvesting Strategies for a Multi-cycle and Multi-pond Shrimp Operation - A Practical Network Model