Document Type : Research Paper


1 Department of Mathematical Sciences, Mazandaran University, Babolsar, Iran

2 Department of Mathematics, Payame Noor University


In the process of milk production, the highest cost relates to animal feed. Based on reports provided by the experts, around seventy percent of dairy livestock costs included feed costs. In order to minimize the total price of livestock feed, according to the limits of feed sources in each region or season, and also the transportation and maintenance costs and ultimately milk price reduction, optimization of the livestock nutrition program is an essential issue. Because of the uncertainty and lack of precision in the optimal food ration done with existing methods based on linear programming, there is a need to use appropriate methods to meet this purpose. Therefore, in this study formulation of completely mixed nutrient diets of dairy cows is done by using a fuzzy linear programming in early lactation. Application of fuzzy optimization method and floating price make it possible to formulate and change the completely mixed diets with adequate safety margins. Therefore, applications of fuzzy methods in feed rations of dairy cattle are recommended to optimize the diets. Obviously, it would be useful to design suitable software, which provides the possibility of using floating prices to set feed rations by the use of fuzzy optimization method.


Main Subjects

Bas, E. (2014). A robust optimization approach to diet problem with overall glycemic load as objective function. Applied Mathematical Modelling, 38, 4926–4940.
Bazaraa, M.S., Jarvis, J.J., & Sherali, H.D. (2005). Linear Programming and Network Flows, 3rd ed., John Wiley and Sons, New York.
Bellman, R.E., & Zadeh, L.A. (1970). Decision making in a fuzzy environment. Management Science, 17, 141–164.
Cadenas, J.M., Pelta, D.A., Pelta, H.R., & Verdegay, J.L. (2004). Application of fuzzy optimization to diet problems in argentienan farms. European Journal of Operational Research, 158, 218–228.      
Castrodeza, C., Lara, P., & Pena, T. (2005). Multi-criteria fractional model for feed formulation: economic, nutritional and environmental criteria. Agricultural Systems, 86, 76–96.
Darvishi SalooKolaei, D., Teimouri Yansari, A., & Nasseri, S.H. (2011). Application of fuzzy optimization in diet formulation. The Journal of Mathematics and Computer Science, 2, 459-468.
Ebrahimnejad, A., & Nasseri, S.H. (2009). Using complementary slackness property to solve linear programming with fuzzy parameters. Fuzzy Information and Engineering, 3, 233-245.
Ebrahimnejad, A., & Nasseri, S.H. (2010). A dual simplex method for bounded linear programmes with fuzzy numbers. International Journal of Mathematics in Operational Research, 2(6), 762-779.
Ebrahimnejad, A., Nasseri, S.H., & Hosseinzadeh Lotfi, F., (2010b). Bounded linear programs with trapezoidal fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18(3), 269-286.
Ebrahimnejad, A., Nasseri, S.H., & Mansourzadeh, S.M., (2011). Bounded primal simplex algorithm for bounded linear programming with fuzzy cost coefficients. International Journal of Operations Research and Information Systems, 2(1), 96-120.
Ebrahimnejad, A., Nasseri, S.H., Hosseinzadeh Lotfi, F., & Soltanifar, M. (2010a). A primal dual method for linear programming problems with fuzzy variables. European Journal of Industrial Engineering, 4(2), 189-209.
Ganesan, K., & Veeramani, P. (2006). Fuzzy linear programming with trapezoidal fuzzy numbers.  Ann. Operations Research, 143, 305-315.
Gupta, R., & Chandan, M. (2013). Use of controlled random search technique for global optimization in animal diet problem. International Journal of Emerging Technology and Advanced Engineering, 3(2), 284-284.
Hsu, K.C. & Wang, F.S. (2013). Fuzzy optimization for detecting enzyme targets of human uric acid metabolism. Bioinformatics, 29(24), 3191–3198.
Iskander, M.G. (2006). Using the weighted max-min approach for stochastic fuzzy goal programming. Computer Math Application, 154, 543-553.
Mahdavi-Amiri, N., & Nasseri, S.H. (2006). Duality in fuzzy number linear programming by use of a certain linear ranking function. Applied Mathematics and Computation, 180, 206-216.
Mahdavi-Amiri, N., & Nasseri, S.H. (2007). Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables. Fuzzy Sets and Systems, 158, 1961-1978.
Mahdavi-Amiri, N., Nasseri, S.H. & Yazdani, A. (2009). Fuzzy primal simplex algorithms for solving fuzzy linear programming problems. Iranian Journal of Operational Research, 1, 68–84.
Maleki, H.R. (2002). Ranking functions and their applications to fuzzy linear programming. Far East J. Math. Sci. (FJMS), 4, 283–301.
Maleki, H.R., Tata, M. & Mashinchi, M. (2000). Linear programming with fuzzy variables. Fuzzy Sets and Systems, 109, 21–33.
Mamat, M., Deraman, S.K., Noor, N.M.M.,  & Mohd, I. (2012). Diet problem and  nutrient requirement using fuzzy  linear programming approaches. Asian Journal of Applied Sciences, 5(1), 52-59.
Mamat, M., Rokhayati, Y., Mohamad, N.N., & Mohd, I. (2011). Optimizing human diet problem with fuzzy price using fuzzy linear approach. Pakistan Journal of Nutrition, 10(6), 594-598.
Marinov, E. & Daumalle, E. (2014). Index matrix interpretation and intuitionistic fuzzy estimation of the diet problem. 18th International Conference on Intuitionistic fuzzy sets system, Notes on Intuitionistic Fuzzy Sets, 20(2), 75–84.
Moraes, L.E., Wilen, J.E., Robinson, P. H., & Fadel, J.G. (2012). A linear programming model to optimize diets in  environmental policy scenario. Journal of Dairy Science, 95, 1267-1282.
Nasseri, S.H. (2008). A new method for solving fuzzy linear programming by solving linear programming. Applied Mathematical Sciences, 2, 2473-2480.
Nasseri, S.H., & Ebrahimnejad, A. (2010a). A fuzzy dual simplex method for fuzzy number linear programming problem. Advances in Fuzzy Sets and Systems, 5(2), 81-95.
Nasseri, S.H., & Ebrahimnejad, A. (2010b). A fuzzy primal simplex algorithm and its application for solving flexible linear programming problems. European Journal of Industrial Engineering, 4(3), 372-389.
Nasseri, S.H., & Mahdavi-Amiri, N. (2009). Some duality results on linear programming problems with symmetric fuzzy numbers. Fuzzy Information and Engineering, 1, 59–66.
National Research Council (NRC). (2001). Nutrient Requirements of Dairy Cattle. The National Academic Press, Washington, DC.
Niemi, J.K., Sevon-Aimonen, M., Pietola, K., & Stalder, K.J. (2010). The value of precision feeding technologies for grow-finish swine. Livestock Science, 129, 13–23.      
Piyaratne, M.K.D.K., Dias, N.G.J., & Attapattu, N.S.B.M. (2012). Linear model based software approach with ideal  amino acid profiles for least-cost poultry ration formulation. Information Technology Journal, 11(7), 788-793.
Pomar, C., Dubeau, F., Letourneau-Montminy, M.P., Boucher, C. and Julien, P.O. (2007). Reducing phosphorus concentration in pig diets by adding an environmental objective to the traditional feed formulation algorithm. Livestock Science, 111, 16–27.
Saxena, P. (2011). Comparison of linear and nonlinear programming techniques for animal diet. Applied Mathematics, 1(2), 106-108.
Tanaka, H., Okuda, T., & Asai, K. (1974). On fuzzy mathematical programming. The Journal of Cybernetics, 3, 37-46.
 Wang, F.S., Wu, W.H., & Hsu, K.C. (2014). Fuzzy optimization in metabolic systems. International Journal of Biological, Food, Veterinary and Agricultural Engineering, 8(7), 661-665.
Yager, R.R. (1981). A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24, 143–161.
Zhang, G., Wu, Y. H., Remias, M., & Lu, J. (2003).  Formulation of fuzzy linear programming problems as four-objective constrained optimization problems. Applied Mathematics and Computation, 139, 383-399.
Zimmermann, H.J. (1978). Fuzzy programming and linear programming with multiple objective functions. Fuzzy Sets and Systems, 45- 55.
Zimmermann, H.J. (1987). Fuzzy set. Decision making and expert systems, Kluwer Academic Publisher.
Zimmermann, H.J. (2001). Fuzzy set theory and applications. 4rd ed. Kluwer Academic Publisher.