Linear Programming Approach to Diet Problem for Black Tiger Shrimp in Shrimp Aquaculture

Transcription

1 Liner Progrmming Approch to Diet Problem for Blck Tiger Shrimp in Shrimp Aquculture Moe Snd Htun, Tin Tin Thein, Pyke Tin University of Computer Studies, Yngon (UCSY), Mynmr Abstrct In this pper, we emine effect of dietry protein qulity on the blck tiger shrimp (Peneus monodon fbricius) nd how food consumption nd diet qulity cn ffect the shrimp growth nd survivl. This model cn predict popultion of the shrimp under shrimp quculture conditions. Simple models for describing nd predicting growth nd feed requirement of blck tiger shrimp under shrimp quculture conditions re presented. The min ide of this reserch is to predict decision mking process for the quculture bioinformtics using proposed models. We present nd disseminte the liner progrmming method, in order to ttin optimized solutions of shrimp quculture problems involving economics nd nutrition. First, we present the formultion of diet t minimum cost using three different formulted feeds. Second we describe mimum growth rte of the shrimp nd lter, we present mimum survivl rte of shrimp under vrious kinds of formulted shrimp feeds. In world with incresingly scrce resources nd every dy more competitive, liner progrmming could be used to serch optimized solutions for quculture problems. Keywords: Liner Progrmming, Blck Tiger Shrimp, Shrimp Feed, Minimum Cost, Mimum Growth Rte, Mimum Survivl Rte. Introduction The quculturists were fcing problem with the lck of good qulity qufeeds s one of the mjor problem. Feed qulity cn be controlled shrimp growth. Growth nd survivl rtes re the fctor of prime importnce in shrimp quculture. Effect of feed nutritionl qulity nd feeding rte cn be predicted well by the model. Growth nd survivl involve the ccretion of body components. The mounts of body components deposited nd the cost of depositing these components re the min fctors determining feed requirement nd wste outputs of shrimp nd other qutic nimls. The use of mthemticl models offers n objective nd prcticl wy of description pttern of growth rte, survivl rte nd predicting cost for food. Effect of feed nutritionl qulity nd feeding rte cn be predicted well by the model, which will provide insight of interctions mong the growth nd growth prmeters of shrimp. Aquculture is one of the fstest growing food production ctivities nd widely prctices in every fesible plce in the world. Mynmr is rich in nturl resources. The Government of its highest level considers quculture generlly nd mrine shrimp frming in prticulr s one of high priority sectors of the economy. However, the crustcen quculture industry is presently in its infncy in Mynmr nd therefore it would benefit from fundmentl reserch tht seeks to emine how the growth nd food conversion efficiency of blck tiger shrimp cn be mimized. The efficient culture of shrimp cn be compred to length. It is dependent on knowledge of nutritionl requirements of the shrimp in order to develop nd deliver suitble diets to the growing shrimps, to minimize food wstge, to promote efficient food conversion nd to mimize growth performnce under culture conditions. A mjor issue in the shrimp frming sector is feed supply. Understnding the growth rte of shrimp is mde on the modifiction of diets nd feeding regimes. The prticulr cost-effective feed being formulted should contin mny individul nutrients tht re essentil for optiml crustcen performnce. The nutritive vlue of dietry protein is governed by the shrimp growth. The influences of diet on growth rte of vrious stges of shrimp hve been studied, however, very little is known bout the protein metbolism of the shrimp. A prticulr key problem tht presents fetures of vgueness, economic relevnce nd humn interction is the so clled diet problem which is of up most importnce in vriety of prcticl situtions. The gol of this problem is to find the chepest combintion of feeds tht will stisfy ll the dily nutritionl requirements of either person or niml. The problem my be formulted s liner progrmming problem where the objective is to minimum cost; mimum growth rte nd mimum survivl rte nd stisfy the constrints. In prcticl pplictions is lmost impossible to hve perfect knowledge of some or ll the dt tking prt into the diet problem t hnd, so it is usul to pproimte those vlues, dt nd/or models in different wys. In prticulr, liner progrmming cn show ll its power in the frmework of decision support systems becuse it is useful for ccurtely modeling the inherent vgueness in the dt tht the user often hs vilble. Also, it my help to find good solutions for problems in which finding n optiml solution is not esy. The problem, we ddress here, the design of diets for blck tiger shrimp in quculture, is fr from being trivil becuse uncertinty nd lck of precision re present in severl spects. For emple the requirements

2 for ech niml vry s function of its weight, ge, se, nd wter conditions (temperture, ph, slinity, hrdness, dissolved oygen, etc.). Becuse the shrimp is not confined, nimls re not mnged individully so, such individul chrcteristics re not vilble. The min im is to design diet which enbles shrimp of certin chrcteristics, to chieve certin level of dily weight gin. So, in principle, it is necessry to determine, first, set of foods nd second, their corresponding mounts (in kg.). Ech food provides certin mount of nutrients nd the diet must verify some requirements on them. The nutrients considered here re crude protein, ft, fiber, sh, moisture, crbohydrte. The most importnt fctors to consider re ph nd slinity. The desirble temperture rnge ws kept 7 9 C. The optimum rnge of se wter ph should be within the rnge of The optimum rnge of slinity should be within the rnge of 5-8 ppt but the shrimp cn survive over rnge of 5-3 ppt. The se wter should hve very low suspended solids preferbly below 0 ppm. Under this scenrio, it mkes sense to consider Diet Problems. In order to solve the former problem on the design of diets for shrimp in quculture, this pper is orgnizes s follows: first we provide literture review for liner progrmming tht hs improved drmticlly to solve rel world problems. We will lso introduce the liner progrmming. Then, the essence of clssicl diet problem models will be described nd then simple method will be presented. The net section shows the result of the model nd conclusion.. Relte Work Opertions Reserch is science designed to provide quntittive tools to decision-mking processes. It comprises set of mthemticl optimiztion nd simultion methods nd models, such s Liner Progrmming, Non-liner Progrmming, Combintory Optimiztion, Theory of Queues, Dynmic Progrmming, Theory of Decisions, etc. Tody, implementing optimized solutions by liner progrmming hs reduced costs by hundreds or even thousnds dollrs in mny middle to lrge-sized compnies in severl industrilized countries.[8] Mny reserchers point out the development of liner progrmming method s one of the most importnt scientific dvnces of the second hlf of the 0 th century. Unfortuntely, few Brzilin compnies use some optimiztion technique in their processes, be them productive or not Liner progrmming hs been used to formulte blnced diets t minimum cost nd complying with set of nutritionl restrictions.[3,6] Moreir et.l.[9] presented nd disseminted the liner progrmming method, in order to ttin optimized solutions of helthcre problems involving Economics nd Nutrition. First, they presented the formultion of diet t minimum cost using only two nutritionl restrictions iming t mking investigtors in the field of helthcre more fmilir with the terms nd potentilities of the method reported. To present mthemticl modeling technique by mens of liner progrmming s n efficient tool to solve problems relted to optimiztion in helthcre. Two pplictions re pproched: formultion of blnced diet t minimum cost nd optiml lloction of resources for set of medicl interventions tht comply with cost nd medicl visit restrictions. In world with incresingly scrce resources nd every dy more competitive, liner progrmming could be used to serch optimized solutions for helthcre problems. The type of lgebric modeling presented in this rticle known s liner progrmming could be considered useful tool to support decision-mking processes in helthcre. In world with incresingly scrce resources nd every dy more competitive, the serch of optimized solutions to replce trditionl methods bsed on common sense nd tril nd error my become n issue of survivl for mny orgniztions. Liner progrmming nlysis is powerful pproch for identifying low-cost nutritionlly dequte diet [], but it hs been only infrequently used for this purpose in humn nutrition [,5]. Briend [] lso illustrted how this technique cn be used to estimte the economic benefits epected from the introduction of different fortified foods using locl food prices. Liner progrmming cn be pplied to identify nutritionlly dequte diet of the lowest cost, since price nd nutrient contents re linerly relted to food weight. Most computer spredsheets now include n esy-to-use solver function tht is suitble for this purpose. This pproch cn lso be used to estimte the effect of introducing food supplement on the miniml cost required to provide nutritionlly dequte diet. It cn lso provide n estimte of the epenses sved by fmilies in reltion to the sums spent by the donor fter the distribution of food supplement. 3. Methods 3.. Liner Progrmming Method Liner progrmming is tool for solving optimiztion problems.[7] Liner progrmming is mthemticl technique tht enbles decision mker to rrive t the optiml solution to problems involving the lloction of scre resources. Typiclly, mny economic nd technicl problems involve mimiztion or minimiztion of certin objective subject to some restrictions. Liner progrmming, in generl, re concerned with the use or lloction of scre resources. Optimiztion models re defined by n objective function composed of set of decision-mking vribles, subject to set of restrictions, nd presented s mthemticl equtions. The objective of optimiztion is to find set of decision-mking vribles tht genertes n optiml vlue for the object function, mimum or minimum vlue depending on the problem, nd complies with set of restrictions imposed by the

3 model. Such restrictions re conditions tht limit the decision-mking vribles nd their reltions to ssume fesible vlues. In liner progrmming models, the objective functions is liner, tht is, it is defined s liner combintion of decision-mking vribles nd set of constnts, restricted to set of liner equlity or inequlity equtions. Therefore, we hve model composed of n objective function, restrictions, decision-mking vribles nd prmeters. We list wht is required in order to construct liner progrmming model: Objective Function: There must be n objective (or gol or trget) the firm or orgniztion wnts to chieve, for emple, mimize profits, minimize cost, mimize totl number of epected potentil customers, nd minimize totl time used, nd so on. Liner Objective Function nd liner Constrints: We must be ble to epress the decision problem incorporting the objective nd restrictions on the decisions using only liner equtions nd liner inequlities. 3 Restrictions nd Decisions: A sign restriction is ssocited with ech vrible. For ny vribles i must be non negtive ( i 0) or tht i my be unrestricted in sign. A solution of problem is clled optiml when the decision-mking vribles ssume vlues tht correspond to the mimum or minimum vlue of the objective function nd complies with ll restrictions of the model. An lgebric representtion of generic formultion of liner progrmming model could be presented s follows: To mimize or minimize the objective function: Z = c i i Z = c + c +...c n n It is subject to restrictions: n n r n n r m + m +... mn n r m non - negtivity : i 0 = ( i,,..., n) Subject to mens tht the vlues of the decision vribles must stisfy ll the constrints nd ll the sign restrictions. 3.. Diet Problem Model The diet problem is one of the first optimiztion problems. The gol of the diet problem is to find n optiml combintion of foods tht will stisfy ll the nutritionl requirements of the shrimp. The problem cn be formulted s liner progrm with n objective nd constrints which require tht dily nutritionl needs be stisfied. To illustrte n ppliction of liner progrmming in diet formultion, we use dt set of blck tiger shrimp food in the Deprtment of Biotechnology of Yngon Technologicl University.[,0] We know tht the nutritionl requirements for the blck tiger shrimp will be epressed s protein, ft, fiber, sh, moisture nd crbohydrte. With regrd to ddress mthemticl version of the problem, we need to define some vribles. Let consider set of foods F = {,,... n }, set of nutrients N = {,,... m }, nd the following vribles: m = the number of nutrients n = the number of foods ij = the number of the i th nutrient of the j th food b i = the minimum number of the i th nutrient required c j = the cost of the j th food j = the number of units the j th food in the diet Hence, in first pproimtion, the model cn be mthemticlly formulted s liner progrmming problem where the objective is to minimize the cost nd mimize the growth rte nd survivl rte while the nutritionl needs (the constrints) re stisfied. Min or M suject to n j= n j= c j j ij j j b i, i =,,..., m 0, j =,,..., n 3... Mthemticl Model for the Diet Problem We formulte mthemticl model tht describes diets problems in generl wy, nd compute minimum cost diets. It is defined by the combintion of food, food nd food 3. Tble 3. summrizes the quntity of ech nutrient vilble in formulted foods nd the shrimp s dily requirement for good growth rte nd survivl rte, s well s the unitry cost of these foods. Tble 3. Nutritionl Vlue of Different Feeds with Cost, survivl rte nd growth rte Nutrient Food Food Food 3 protein Ft Fiber Ash Moisture Crbonhydrte Cost/unit Growth rte Survivl rte

4 We used bove tble to solve my reserch dt. We lso describe the lgorithm of liner progrmming used to get optimiztion result. The objective is to minimize the totl diet cost. Decision-mking vribles: = number of nutrient in food = number of nutrient in food 3 = number of nutrient in food 3 Objective function (Z): The function to be minimized totl diet cost is the objective function of this problem. It is defined by the combintion of food, food nd food 3. Their unit costs re 77 cost/unit, 709 cost/unit nd 754 cost/unit respectively. The cost function is liner function of, nd 3 tht is: z = Restrictions: The totl quntity of protein in this diet should be equl or greter thn The food formultion should provide t lest 3.65 of ft,.3 of fiber, 5.3 of sh, 7. of moisture nd 7.84 of crbohydrte. We could not tke into ccount negtive quntities of food; thus, nd 3 should be non-negtive quntities ( 0; 0; 3 0) We formulte mthemticl model tht describes diets problems in generl wy, nd compute minimum cost diets. It is defined by the combintion of food, food nd food 3. To minimize cost: z = it is subject to: (protein restriction) (ft restriction) (fiber restriction) (sh restriction) (moisture restriction) (crbohydrte restriction) 0; 0; 3 0 (non-negtivity restriction) Other problems re growth rte nd survivl rte. The function to be mimized growth rte nd survivl rte re the objective functions. There re severl methods of formulting feeds, nmely the Algebric equtions, Liner Progrmming nd Person s Squre methods. The most commonly used method in formulting supplementl feeds is by the liner progrmming. To mimize growth rte: Z = it is subject to : (protein restriction) (ft restriction) (fiber restriction) (sh restriction) (moisture restriction) (crbohydrte restriction) 0; 0; 3 0 (non-negtivity restriction) To mimize survivl rte: Z = it is subject to : (protein restriction) (ft restriction) (fiber restriction) (sh restriction) (moisture restriction) (crbohydrte restriction) 0; 0; 3 0 (non-negtivity restriction) 3.3 Simple Method Shrimp feeding Problems cn be set up s mthemticl optimiztion problem. In Opertions Reserch mny methods re used to find solutions for decision problems. Problems cn be set up s liner progrm. A specil method for solving problems in the liner plnning clcultion is the simple lgorithm. Step : Convert the liner progrmming to stndrd form. Step : Obtin n initil bsic fesible solution. Step 3: () If ll nonbsic vribles hve positive coefficients in row 0, the current bsic fesible solution is the unique optiml solution to the liner progrmming. Stop. (b) If t lest one nonbsic vrible hs zero coefficient in row 0, while ll other nonbsic vribles hve positive coefficients, the current bsic fesible solution is optiml. However, the

5 liner progrmming my hve multiple optiml solutions. Use the nonbsic vrible with the zero coefficients s the entering vrible nd continue with Step 5 to determine other optiml solutions. Step 4: Choose the nonbsic vrible with the most negtive coefficient in row 0 to enter the bsic; this vrible is clled the entering vrible. Step 5: () If the entering vrible hs nonpositive coefficient in ech constrint, the liner progrmming is unbounded. Stop. (b) For ech constrint in which the entering vrible hs positive coefficient, compute the following rtio: Right hnd side of row Coefficient of entering vrible in row A row ttining the smllest vlue of this rtio is the winner of the rtio test; the bsic vrible for this row is the deprting vrible. Step 6: Mke the entering vrible bsic vrible in the constrint tht wins the rtio test. Go bck to Step Results 4. Eperimentl results Bsed on Simple Method 4. Grphic solution for the diet problem When we del with decision-mking vribles (three foods), grphic representtion is possible nd convenient for didctic purposes. The results cn be seen s the following grph. The evlution cn be used to choose right nd proper decision which is mking potentil to enhnce the benefits of the system. In Figure 4., the growth rte of the shrimp bsed on the input prmeters is shown nd the survivl rte is ddition depicted in Figure 4.3. The cost sttistics condition estimtion on those foods is presented in the form of grph s shown in Figure 4.4 groth rte feed type growth rte Figure 4. Growth Rte of the Shrimp Bsed on the Input Prmeters In this section, we present the some prcticl results of liner progrmming simple method by mens of rel dt sets. The implement simultion depiction of our system is shown in Figure 4.. survivl rte survivl rte Simple method food type User Figure 4.3 Survivl Rte of the Shrimp Bsed on the Input Prmeters food type tet user Output Result - Survivl rte - Growth rte - Cost Figure 4. Fundmentl Architecture of the System cost food type cost Figure 4.4 Cost of the Shrimp Bsed on the Input Prmeters

6 4.. Anlyticl solution for the diet problem Our clcultion is bsed on simple method lgorithm. The Simple Method is n lgorithm creted to lgebriclly obtin solution. An lgorithm is set of rules tht must be followed step by step, so tht, in the end, the desired result is ttined. The Simple Method ws creted by George Dntzing nd other scientists of the Americn Air Force Deprtment, in 947. [7] The optiml solution obtined for this problem clculted by simple method. This solution provides mimum growth rte nd mimum survivl rte. This solution indictes tht of food, of food, of food 3 for of cost nd of food, of food, of food 3 for of growth rte nd of food, of food, of food 3 for of survivl rte. In cses of si or more decision-mking vribles, the only lterntive is n nlyticl solution by Simple Method. The demnd for efficient decisions in quculture gives opportunity to ppliction of optimiztion techniques in problems relted to resource lloction, which could be complementry tool to economic evlution models. The diet proposed in this model provides reduced number of restrictions, decisionmking vribles nd the type of restriction used since it hs the didctic purpose of demonstrting tht this type of lgebric modeling is efficient. This model could benefit from formultion of blnced diets t minimum cost, prepred bsed on foods vilble nd/or ffordble, thus complying with dozens or hundreds of restrictions. Potentil pplictions of optimiztion methods my include ssessment of economic impct of severl therpies by mens of evlution of product prices. For instnce, optimized prices of product components could be ssessed considering some restrictions, such s epenses with reserch nd development, cost of lterntive tretments, mrketing strtegies nd estimtes for sles projection. Mthemticl modeling using liner progrmming my be pplied to problems relted to optimized resource lloction in quculture. 5. Conclusion The type of lgebric modeling presented in this pper known s liner progrmming could be considered useful tool to support decision-mking processes in quculture. In world with incresingly scre resources nd every dy more competitive, the serch of optimized solutions to replce trditionl methods bsed on common sense nd tril nd error my become n issue of survivl of mny orgniztions. The objective could be to minimize the cost of tretment, s well s to define the components for diet formultion nd to mimize the growth rte nd survivl rte of the blck tiger shrimp. The finl im of the diets here considered is to promote the fttening nd survivl of the shrimp. Hving the option of performing n evlution of the nutritionl chrcteristics of the diets, besides the cost, it is very importnt point becuse enble the decisor to choose the diet blncing two opposed items: cost nd nutritionl chrcteristics. The opportunity of mking mnul djustments of the mounts of foods, enble the epert to compre its own solutions with those provided by the system, or to use them s guide to construct new solutions being better with respect to chrcteristics not reflected under the current modeliztion of the problem. Reference [] K. M. Aye, Shrimp Feed Formultion For Post Lrve Stges of Peneus Monodon Using Squid Mel, Mster s Thesis, Yngon Technologicl University, Mynmr, April, 003. [] A. Briend, N. Drmon, Determining Limiting Nutrients by Liner Progrmming: New Approch to Predict Insufficient Intkes from Complementry Foods, Peditrics 000, vol.06, no.5, pp.88 89, November 000. [3] C. Colvit nd R. D Orsi, Liner Progrmming nd Peditric Dietetics, Br J Nutr, vol.64, pp.307-7, 990. [4] G. B. Dntzig, Origins of the Simple Method In: Nsh SG, editor. A history of scientific computing, New York: ACM Press, 990. [5] J. Foytik, Very Low-Cost Nutritious Diet Plns Designed by Liner Progrmming, J Nutr Educ 98, vol.3, pp.6369, 98. [6] S. Henson, Liner Progrmming Anlysis of Constrints Upon Humn Diets, J Agric Econ, vol. 4, pp , 99. [7] F. S. Hillier, G. J. Liebermn, Introduction to Opertions Reserch, Seventh Edition, McGrw-Hill Press, 00. [8] G. J. Liebermn, F. S. Hillier, Introduction to Mthemticl Progrmming, Second Edition, New York, McGrw-Hill Press, 99. [9] F. R. Moreir, Liner Progrmming Applied to Helthcre Problems, Clinicl Reserch Center of Hospitl Isrelit Albert Einstein, Sttisticin of the Clinicl Reserch Center, Teching nd Reserch Institute, November 0, 003. [0] V. E. Smith, Liner Progrmming Models for the Determintion of Pltble Humn Diets, J Frm Econ 959, vol.3, pp.7 83, 959. [] T. M. Thein, Comprtive Study on the Growth nd Survivl of Peneus Monodon Fbricius Post Lrve Using Different Formulted Diets, Mster s Thesis, Yngon Technologicl University, Mynmr, April, 00. [] N. Toft, Elements of Decision Support Systems in Pig Production, Ph.D Thesis, The Royl Veterinry nd Agriculturl University, Copenhgen, Din Reserch Report No.90, 000.