Bulletin of Mathematical Biology

, Volume 76, Issue 5, pp 997–1016 | Cite as

Mathematical Programming Models for Determining the Optimal Location of Beehives

  • Maica Krizna A. Gavina
  • Jomar F. Rabajante
  • Cleofas R. Cervancia
Original Article

Abstract

Farmers frequently decide where to locate the colonies of their domesticated eusocial bees, especially given the following mutually exclusive scenarios: (i) there are limited nectar and pollen sources within the vicinity of the apiary that cause competition among foragers; and (ii) there are fewer pollinators compared to the number of inflorescence that may lead to suboptimal pollination of crops. We hypothesize that optimally distributing the beehives in the apiary can help address the two scenarios stated above. In this paper, we develop quantitative models (specifically using linear programming) for addressing the two given scenarios. We formulate models involving the following factors: (i) fuzzy preference of the beekeeper; (ii) number of available colonies; (iii) unknown-but-bounded strength of colonies; (iv) probabilistic carrying capacity of the plant clusters; and (v) spatial orientation of the apiary.

Keywords

Beekeeping Pollination management Apis mellifera Stingless bees Bumblebees Crop pollination 

References

  1. Aizen, M. A., & Harder, L. D. (2009). The global stock of domesticated honey bees is growing slower than agricultural demand for pollination. Curr. Biol., 19, 915–918. CrossRefGoogle Scholar
  2. Aizen, M. A., Garibaldi, L. A., Cunningham, S. A., & Klein, A. M. (2008). Long-term global trends in crop yield and production reveal no current pollination shortage but increasing pollinator dependency. Curr. Biol., 18, 1572–1575. CrossRefGoogle Scholar
  3. Allen-Wardell, G., et al. (1998). The potential consequences of pollinator declines on the conservation of biodiversity and stability of food crop yields. Conserv. Biol., 12(1), 8–17. CrossRefGoogle Scholar
  4. Almazol, A. E., & Cervancia, C. R. (2010). Foraging behaviour of Xylocopa spp., Apis dorsata and Apis cerana on two mangrove species (Aegiceras floridum Roem & Shults and Scyphiphora hydrophyllacea Gaertn F.) in Pagbilao Mangrove, Quezon Province, Philippines. In Proceedings of the 10th Asian Apicultural Association Conference and Api Expo, Busan, South Korea (pp. 4–7). Google Scholar
  5. Association, B. B. (2006). Choosing an Apiary site. British Beekeepers Association Advisory Leaflet B11 (3rd ed.). Warwickshire: British Beekeepers Association. Google Scholar
  6. Association, B. B. (2013). Keeping bees—apiary set up. Available from http://barnsleybeekeepers.org.uk/apiary.html. Accessed June 24.
  7. Awasthi, A., Chauhan, S. S., & Goyal, S. K. (2011). A multi-criteria decision making approach for location planning for urban distribution centers under uncertainty. Math. Comput. Model., 53, 98–109. MathSciNetCrossRefMATHGoogle Scholar
  8. Behrens, D., Forsgren, E., Fries, I., & Moritz, R. F. A. (2007). Infection of drone larvae (Apis mellifera) with American foulbrood. Apidologie, 38, 281–288. CrossRefGoogle Scholar
  9. Beil, M., Horn, H., & Schwabe, A. (2008). Analysis of pollen loads in a wild bee community (Hymenoptera: Apidae)—a method for elucidating habitat use and foraging distances. Apidologie, 39, 456–467. CrossRefGoogle Scholar
  10. Bisschop, J. (2012). AIMMS optimization modeling. Haarlem: Paragon Decision Technology. Google Scholar
  11. Bosaing, A. A. D., Rabajante, J. F., & De Lara, M. L. D. (2012). Assignment problems with weighted and nonweighted neighborhood constraints in 36, 44 and 63 tilings. Southeast. Asian J. Sci., 1(1), 55–75. Google Scholar
  12. Bradbear, N. (2009). Bees and their role in forest livelihoods: a guide to the services provided by bees and the sustainable harvesting, processing and marketing of their products. Rome: Food and Agriculture Organization of the United Nations. Google Scholar
  13. Brittain, C., Williams, N., Kremen, C., & Klein, A. M. (2013). Synergistic effects of non-Apis bees and honey bees for pollination services. Proc. R. Soc. B, 280(1754), 20122767. CrossRefGoogle Scholar
  14. Brosi, B. J., Armsworth, P. R., & Daily, G. C. (2008). Optimal design of agricultural landscapes for pollination services. Conserv. Lett., 1(1), 27–36. CrossRefGoogle Scholar
  15. Cervancia, C. R., & Bergonia, E. A. (1991). Insect pollination of cucumber (Cucumis sativus L) in the Philippines. Acta Hortic., 228, 278–281. CrossRefGoogle Scholar
  16. Cervancia, C. R., & Forbes, M. F. (1993). Pollination of pechay (Brassica pekinensis Rupr) and radish (Raphanus sativus L). Philipp. J. Sci., 122(1), 129–132. Google Scholar
  17. Charikar, M., Khuller, S., Mount, D. M., & Narasimhan, G. (2001). Algorithms for facility location problems with outliers. In Proceedings of the 12th ACM–SIAM symposium on discrete algorithms, Washington DC (pp. 642–651). Google Scholar
  18. Church, R. L., & Revelle, C. S. (1974). The maximal covering location problem. Pap. Reg. Sci. Assoc., 32, 101–118. CrossRefGoogle Scholar
  19. Dat, L. Q., Yu, V. F., & Chou, S.-Y. (2012). An improved ranking method for fuzzy numbers based on the centroid-index. Int. J. Fuzzy Syst., 14(3), 413–419. MathSciNetGoogle Scholar
  20. De la Rua, P., Jaffe, R., Dall’olio, R., Muñoz, I., & Serrano, J. (2009). Biodiversity, conservation and current threats to European honeybees. Apidologie, 40, 263–284. CrossRefGoogle Scholar
  21. Delaplane, K. S., van der Steen, J., & Guzman-Novoa, E. (2013). Standard methods for estimating strength parameters of Apis mellifera colonies. In V. Dietemann, J. D. Ellis, & P. Neumann (Eds.), The COLOSS BEEBOOK, Vol. I: Standard methods for Apis mellifera research. J. Apicultural Res., 52(1), 52.1.03. Google Scholar
  22. Deyto, R. C., & Cervancia, C. R. (2009). Floral biology and pollination of ampalaya Momordica charantia L. Philipp. Agric. Sci., 92(1), 8–18. Google Scholar
  23. Di Prisco, G., Cavaliere, V., Annoscia, D., Varricchio, P., Caprio, E., Nazzi, F., Gargiulo, G., & Pennacchio, F. (2013). Neonicotinoid clothianidin adversely affects insect immunity and promotes replication of a viral pathogen in honey bees. PNAS, 110(46), 18466–18471. CrossRefGoogle Scholar
  24. Duffield, G. E., Gibson, R. C., Gilhooly, P. M., Hesse, A. J., Inkley, C. R., Gilbert, F. S., & Barnard, C. J. (1993). Choice of flowers by foraging honey bees (Apis mellifera): possible morphological cues. Ecol. Entomol., 18, 191–197. CrossRefGoogle Scholar
  25. Esteves, R. J. P., Villadelrey, M. C., & Rabajante, J. F. (2010). Determining the optimal distribution of bee colony locations to avoid overpopulation using mixed integer programming. J. Nat. Stud., 9(1), 79–82. Google Scholar
  26. Fajardo, A. C., Medina, J. R., Opina, O. S., & Cervancia, C. R. (2008). Insect pollinators and floral visitors of mango, Mangifera indica var. “carabao”. Philipp. Agric. Sci., 91(4), 372–382. Google Scholar
  27. Farahani, R. Z., & Hekmatfar, M. (2009). Facility location: concepts, models, algorithms and case studies. Berlin: Springer. CrossRefGoogle Scholar
  28. Free, J. (1993). Insect pollination of crops. London/New York: Academic Press. Google Scholar
  29. Gallai, N., Salles, J. M., Settele, J., & Vaissiere, B. E. (2009). Economic valuation of the vulnerability of world agriculture confronted with pollinator decline. Ecol. Econ., 68, 810–821. CrossRefGoogle Scholar
  30. Genersch, E., von der Ohe, W., Kaatz, H., Schroeder, A., Otten, C., Buchler, R., Berg, S., Ritter, W., Muhlen, W., Gisder, S., Meixner, M., Liebig, G., & Rosenkranz, P. (2010). The German bee monitoring project: a long term study to understand periodically high winter losses of honey bee colonies. Apidologie, 41, 332–352. CrossRefGoogle Scholar
  31. Ghazoul, J. (2005). Buzziness as usual? Questioning the global pollination crisis. Trends Ecol. Evol., 20(7), 367–373. CrossRefGoogle Scholar
  32. Guan, Z., Jin, Z., & Zou, B. (2007). A multi-objective mixed-integer stochastic programming model for the vendor selection problem under multi-product purchases. Inf. Manag. Sci., 18(3), 241–252. MathSciNetMATHGoogle Scholar
  33. Guzman-Novoa, E., Eccles, L., Calvete, Y., McGowan, J., Kelly, P. G., & Correa-Benitez, A. (2010). Varroa destructor is the main culprit for the death and reduced populations of overwintered honey bee (Apis mellifera) colonies in Ontario, Canada. Apidologie, 41, 443–450. CrossRefGoogle Scholar
  34. Herrera, F., & Verdegay, J. L. (1995). Three models of fuzzy integer linear programming. Eur. J. Oper. Res., 83, 581–593. CrossRefMATHGoogle Scholar
  35. Jadczak, A. (1994). Placement of honey bee colonies used for blueberry pollination. Maine: University of Maine. Google Scholar
  36. Kevan, P. G. (1999). Pollinators as bioindicators of the state of the environment: species, activity and diversity. Agric. Ecosyst. Environ., 74, 373–393. CrossRefGoogle Scholar
  37. Klein, A. M., Steffan-Dewenter, I., & Tscharntke, T. (2002). Bee pollination and fruit set of Coffea arabica and C. canephora (Rubiaceae). Am. J. Bot., 90(1), 153–157. CrossRefGoogle Scholar
  38. Klein, A.-M., Vaissiere, B. E., Cane, J. H., Steffan-Dewenter, I., Cunningham, S. A., Kremen, C., & Tscharntke, T. (2007). Importance of pollinators in changing landscapes for world crops. Proc. R. Soc. Lond. B, Biol. Sci., 274(1608), 303–313. CrossRefGoogle Scholar
  39. Kuhn-Neto, B., Contrera, F. A. L., Castro, M. S., & Nieh, J. C. (2009). Long distance foraging and recruitment by a stingless bee Melipona mandacaia. Apidologie, 40, 472–480. CrossRefGoogle Scholar
  40. Le Conte, Y., Ellis, M., & Ritter, W. (2010). Varroa mites and honey bee health: can Varroa explain part of the colony losses? Apidologie, 41, 353–363. CrossRefGoogle Scholar
  41. Li, Y. P., Huang, G. H., Huang, Y. F., & Zhou, H. D. (2009). A multistage fuzzy-stochastic programming model for supporting sustainable water-resources allocation and management. Environ. Model. Softw., 24, 786–797. CrossRefGoogle Scholar
  42. Liu, B. (2009). Theory and practice of uncertain programming (2nd ed.). Berlin: Springer. CrossRefMATHGoogle Scholar
  43. Lowore, J., & Bradbear, N. (2012). Extensive beekeeping. Bees Dev. J., 103, 3–5. Google Scholar
  44. Mader, E., Spivak, M., & Evans, E. (2010). Managing alternative pollinators: a handbook for beekeepers, growers and conservationists (SARE Handbook 11). Maryland: Sustainable Agriculture Research and Education and New York: Natural Resource, Agriculture, and Engineering Service. Google Scholar
  45. Maini, S., Medrzycki, P., & Porrini, C. (2010). The puzzle of honey bee losses: a brief review. Bull. Insectol., 63(1), 153–160. Google Scholar
  46. Manila-Fajardo, A. C. (2011). Pollination biology of coffea liberica W. Bull ex Hiern var. liberica in Lipa City, Philippines. PhD thesis, University of the Philippines Los Baños. Google Scholar
  47. Megiddo, N., Zemel, E., & Hakimi, S. L. (1983). The maximum coverage location problem. SIAM J. Algebr. Discrete Methods, 4(2), 253–261. MathSciNetCrossRefMATHGoogle Scholar
  48. Menz, M. H. M., Philips, R. D., Winfree, R., Kremen, C., Aizen, M. A., Johnson, S. D., & Dixon, K. W. (2011). Reconnecting plants and pollinators: challenges in the restoration of pollination mutualisms. Trends Plant Sci., 16(1), 4–12. CrossRefGoogle Scholar
  49. Morse, R. A., & Calderone, N. W. (2000). The value of honey bees as pollinators of US crops in 2000. In Bee culture magazine, Ohio: A.I. Root Company. Google Scholar
  50. Murray, T. E., Kuhlmann, M., & Potts, S. G. (2009). Conservation ecology of bees: populations, species and communities. Apidologie, 40, 211–236. CrossRefGoogle Scholar
  51. Oldroyd, B. P., & Nanork, P. (2009). Conservation of Asian honey bees. Apidologie, 40, 296–312. CrossRefGoogle Scholar
  52. Park, M. G., Orr, M. C., & Danforth, B. N. (2010). The role of native bees in apple pollination. New York Fruit Q., 18(1), 21–25. Google Scholar
  53. Pettis, J. S., Lichtenberg, E. M., Andree, M., Stitzinger, J., Rose, R., & van Engelsdorp, D. (2013). Crop pollination exposes honey bees to pesticides which alters their susceptibility to the gut pathogen nosema ceranae. PLoS ONE, 8(7), e70182. CrossRefGoogle Scholar
  54. Potts, S. G., Vulliamy, B., Dafni, A., Ne’eman, G., & Willmer, P. (2003). Linking bees and flowers: how do floral communities structure pollinator communities? Ecology, 84, 2628–2642. CrossRefGoogle Scholar
  55. Potts, S. G., Biesmeijer, J. C., Kremen, C., Neumann, P., Schweiger, O., & Kunin, W. E. (2010). Global pollinator declines: trends, impacts and drivers. Trends Ecol. Evol., 25(6), 345–353. CrossRefGoogle Scholar
  56. Raju, A. J. S., & Karyamsetty, H. J. (2008). Reproductive ecology of mangrove trees Ceriops decandra (Griff.) Ding Hou and Ceriops tagal (Perr.) C.B. Robinson (Rhizophoraceae). Acta Bot. Croat., 67(2), 201–208. Google Scholar
  57. Ramalho, M., Kleinert-Giovannini, A., & Imperatriz-Fonseca, V. L. (1989). Utilization of floral resources by species of Melipona (Apidae, Meliponinae): floral preferences. Apidologie, 20, 185–195. CrossRefGoogle Scholar
  58. Roubik, D. W., Yanega, D., Aluja, S. M., Buchmann, S. L., & Inouye, D. W. (1995). On optimal nectar foraging by some tropical bees (Hymenoptera: Apidae). Apidologie, 26, 197–211. CrossRefGoogle Scholar
  59. Sagili, R. R., & Burgett, D. M. (2011). Evaluating honey bee colonies for pollination: a guide for commercial growers and beekeepers. PNW 623. Pacific Northwest Extension Publication. Google Scholar
  60. Slaa, E. J., Tack, A. J. M., & Sommeijer, M. J. (2003). The effect of intrinsic and extrinsic factors on flower constancy in stingless bees. Apidologie, 34, 457–468. CrossRefGoogle Scholar
  61. Slaa, E. J., Chaves, L. A. S., Malagodi-Braga, K. S., & Hofstede, F. E. (2006). Stingless bees in applied pollination: practice and perspectives. Apidologie, 37, 293–315. CrossRefGoogle Scholar
  62. Stephens, D. W., & Stevens, J. R. (2001). A simple spatially explicit ideal-free distribution: a model and an experiment. Behav. Ecol. Sociobiol., 49, 220–234. CrossRefGoogle Scholar
  63. Taha, H. A. (2010). Operations research: an introduction (9th ed.). New Jersey: Prentice Hall. MATHGoogle Scholar
  64. Tambaoan, R. S., Rabajante, J. F., Esteves, R. J. P., & Villadelrey, M. C. (2011). Prediction of migration path of a colony of bounded-rational species foraging on patchily distributed resources. Adv. Stud. Biol., 3(7), 333–345. Google Scholar
  65. Thode, H. C. (2002). Testing for normality. New York: Marcel Dekker/CRC Press. CrossRefMATHGoogle Scholar
  66. Triantaphyllou, E., Shu, B., Nieto Sanchez, S., & Ray, T. (1998). Multi-criteria decision making: an operations research approach. In J. G. Webster (Ed.), Encyclopedia of electrical and electronics engineering (pp. 175–186). New York: Wiley. Google Scholar
  67. Tubay, J. M., Panopio, R. G., & Mendoza, G. A. (2010). A fuzzy multiple objective linear programming model for the optimal allocation of feedstock for bioethanol production. UPLB J., 8, 159–179. Google Scholar
  68. van Engelsdorp, D., & Meixner, M. D. (2010). A historical review of managed honey bee populations in Europe and the United States and the factors that may affect them. J. Invertebr. Pathol., 103, S80–S95. CrossRefGoogle Scholar
  69. van Engelsdorp, D., Hayes, J. Jr., Underwood, R. M., & Pettis, J. (2008). A survey of honey bee colony losses in the US, Fall 2007 to Spring 2008. PLoS ONE, 3(12), e4071. CrossRefGoogle Scholar
  70. van Nieuwstadt, M. G. L., & Ruano Iraheta, C. E. (1996). Relation between size and foraging range in stingless bees (Apidae, Meliponinae). Apidologie, 27, 219–228. CrossRefGoogle Scholar
  71. Wang, Y.-M., Yanga, J.-B., Xu, D.-L., & Chin, K.-S. (2006). On the centroids of fuzzy numbers. Fuzzy Sets Syst., 157, 919–926. CrossRefMathSciNetMATHGoogle Scholar
  72. Williams, G. R., Tarpy, D. R., van Engelsdorp, D., Chauzat, M.-P., Cox-Foster, D. L., Delaplane, K. S., Neumann, P., Pettis, J. S., Rogers, R. E. L., & Shutler, D. (2010). Colony collapse disorder in context. BioEssays, 32(10), 845–846. CrossRefGoogle Scholar
  73. Woodcock, T. S. (2012). Pollination in the agricultural landscape: best management practices for crop pollination. Ontario: Canadian Pollination Initiative (NSERC-CANPOLIN), University of Guelph. Google Scholar

Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  • Maica Krizna A. Gavina
    • 1
  • Jomar F. Rabajante
    • 1
    • 2
  • Cleofas R. Cervancia
    • 2
    • 3
  1. 1.Institute of Mathematical Sciences and PhysicsUniversity of the Philippines Los BañosLagunaPhilippines
  2. 2.University of the Philippines Los Baños Bee ProgramUniversity of the Philippines Los BañosLagunaPhilippines
  3. 3.Institute of Biological SciencesUniversity of the Philippines Los BañosLagunaPhilippines

Personalised recommendations