A New Uncertain Programming Model for Grain Supply Chain Design

INFORMATION Volume 5, Number, pp.-8 ISSN 343-4500 c 0 International Information Institute A New Uncertain Programming Model for Grain Supply Chain Design Sibo Ding School of Management, Henan University of Technology, Zhengzhou 45000, China d4b006@yahoo.com.cn Abstract Grain supply chain design is one of the important and challenging problems in the field of agri-food supply chain design. This paper presents an uncertain grain supply chain model in which the quantity of grain sold by farmers, setup costs of factories and stores are uncertain variables. The model can be transformed into a deterministic form by using operational law of uncertainty theory. Moreover, a hybrid intelligent algorithm to solve this model is given by integrating 99-method and differential evolution algorithm. Finally, a numerical example is presented to illustrate the effectiveness of hybrid intelligent algorithm. Keywords: grain supply chain, uncertain programming, uncertainty theory, differential evolution algorithm Introduction Grain supply chain design plays an important role in grain supply chain management. Several researchers have studied the corresponding problems under stochastic environment. Kaiser et al. (993) developed discrete stochastic sequential programming to examine potential economic and agronomic impacts of gradual climate warming at the farm level. Schilizzi and Kingwell (999) investigated the stochastic bioeconomic farming system model of an uncertain dry land agricultural system. Darby-Dowman et al. (000) presented a twostage stochastic programming model for the problem of determining optimal planting plans. In many cases, probability distributions of some random variables are difficult or impossible to obtain due to the lac of data. Some parameters are supposed to be fuzzy variables via experiences and udgments of experts. Biswas and Pal (005) presented a fuzzy goal programming for modeling and solving land-use planning problems. However, some information and nowledge represented by human language behave neither lie randomness nor fuzziness. In this paper, we propose a new grain supply chain model based on uncertainty theory. The rest of the paper is structured as follows. Some basic concepts and properties of uncertain variables are given in Section. Section 3 presents a new uncertain grain supply chain model, and gives its corresponding deterministic form. To solve the proposed model, Section 4 introduces a hybrid intelligent algorithm which integrates 99-method and different evolution algorithm. The last section contains some conclusions. Preliminary In order to deal with human uncertainty, an uncertainty theory was founded by Liu (007) and refined by Liu (0) based on normality, duality, subadditivity and product axioms. Many researchers have done a lot of significant wor in this area. Wei and Xu (00) proved uncertain Liapunov inequality and discussed the continuity of expected value operator. Yang (0) proved a moments and tails inequality for uncertain variables. Peng and Iwamura (0) gave a relation between set functions and uncertain measures, and proved a sufficient and necessary condition for uncertain measures. Besides, uncertainty theory has been extended to the fields of uncertain programming (Liu 009, Gao 0), uncertain inference (Liu 00b), uncertain logic (Liu 0a, Chen and Ralescu 0) and uncertain delphi method (Wang, Gao and Guo 0). In this section, we introduce some foundational concepts and properties on uncertainty theory. Definition (Liu 007) Let ξ be an uncertain variable. Then the uncertainty distribution Φ of ξ is defined by Φ(x) = M{ξ x}

A NEW UNCERTAIN PROGRAMMING MODEL FOR GRAIN SUPPLY CHAIN DESIGN for any real numbers x. Definition (Liu 0) Let ξ be an uncertain variable with regular uncertainty distribution Φ(x).Then the inverse function Φ (x) is called inverse uncertainty distribution of x. Example The inverse uncertainty distribution of normal uncertain variable σ 3 α Φ (α) = e + ln. α N(e, σ) is N Theorem (Liu 0) Let ξ and ξ be independent normal uncertain variables (e, σ ) and respectively. Then the ξ + ξ is also a normal uncertain variable (e + e, σ + σ ), i.e., N N(e, σ ), N(e, σ ) + N(e, σ ) = N(e + e, σ + σ ). The product of a normal uncertain variable N(e, σ ) and a scalar number > 0 is also a normal uncertain variable N(e, σ ), i.e., N(e, σ ) = N(e, σ ). 3 Problem Description Assume farmers sell grain to factories. Factories can t ensure all grain is accepted, and then set λ as the service level which means λ percent of the grain is accepted. We assume all accepted grain is transported to stores. On the other hand, in order to get grain from farmers, factories are built within the maximum allowable distance l from farmers. Fig. shows a three-echelon grain supply chain. Figure : Setch map of grain supply chain The grain supply chain design problem is to determine the amount of grain accepted by factories, and choose the sites of factories and stores from the potential sites. Prior to developing the networ, we list model parameters and decision variables as follows. 3. Index Set i: index for farmers, i I; : index for factories, J; : index for stores, K; 3. Model Parameters ζi : uncertain quantity of grain sold by farmer i; ξ : uncertain setup cost of opening factory ; γ : uncertain setup cost of opening store ; h : unit handling cost of factory ; 3

SIBO DING d : distance from factory to store ; c: unit transportation cost from factory to store ; l: maximum allowable distance from farmer i to factory ; λ: service level; α: cost confidence level; β : service confidence level; 3.3 Decision Variables x : the amount of grain transported from factory to store ; {, if factory services farmer i Y i = 0, otherwise; {, if factory is opened F O = 0, otherwise; {, if store is opened SO = 0, otherwise. 3.4 Uncertain Grain Supply Chain Design Model The total cost Θ of grain supply chain is as follows: Θ = F O ξ + SO γ + h Y i ζ i + i SO c x d. () The total cost consists of four parts including setup costs of factories and stores, handling cost and transportation cost. The first term is the setup cost of factories. The second term is the setup cost of stores. The third term is handling cost. The fourth term is transportation cost from factories to stores. In fact, quantity of grain sold by farmer ζ i, cost of opening factory ξ i and store γ are not fixed. In this case, we may assume they are uncertain variables. As a function of ζ i, ξ and γ, Θ is also an uncertain variable. Then we have the following model, min C 0 subect to: M{Θ C 0 } α () M{λζ i Y i x i } β, i I (3) Y i =, i I (4) Y i x i = x, J i (5) d i Y i l, i I, J (6) F O (7) SO (8) x 0, J, K (9) Y i, F O, SO {0, }, i I, J, K. (0) Obective and constraint () mean that the budget target is C 0 and the chance that the cost does not exceed the budget is ept above a certain level α. Constraint (3) ensures that the total quantity of grain accepted by factory must satisfy service level λ with the confidence level β. Constraint (4) assures that a 4

A NEW UNCERTAIN PROGRAMMING MODEL FOR GRAIN SUPPLY CHAIN DESIGN farmer is assigned to a factory. Constraint (5) maes the incoming flow equal to the outgoing at a factory. Constraint (6) implies that each factory should be located within a certain allowable proximity of farmers. Constraint (7) and (8) maintain at least one factory and one store. Constraint (9) preserves the non-negativity of decision variables x. Constraint (0) assures the binary nature of decision variables Y i, F O, SO. If ζ i, ξ and γ are independent and have the same types of membership functions, then we can get the analytic form of the model. Theorem Suppose that ζ i, ξ and γ are independent uncertain variables with normal uncertainty distributions N(e i, σ i ), N(e, σ ) and N(e, σ ) for each i, and, respectively. Then the model is equivalent to the following deterministic programming problem: subect to: min e + 3σ λe i + λ 3σ i constraints (4)-(0) ( ) α ln α ( ) βi ln β i Y i x i, J where e = F O e + SO e + h Y i e i + i σ = F O σ + SO σ + SO c x d, h Y i σ i. i Proof: Since ζ i, ξ and γ are normal uncertain variables, using Theorem, we obtain that the total cost Θ is a normal uncertain variable with normal uncertainty distribution N(e, σ ). According to example, the obective of model is equivalent to minimizing e + 3σ ( ) α ln α In the same way, constrain () is equivalent to the following form λe i + λ 3σ i ( ) βi ln β i Y i x i. The proof is completed. Theorem indicates the model may be translated into a deterministic model under some particular conditions. While these conditions cannot be satisfied, the model is difficult or impossible to be translated into a deterministic programming problem. Therefore, the general form of the model has to be solved by the hybrid intelligent algorithm. 4 Solution Methodology In this section, a 99-method is integrated with differential evolution algorithm to produce a hybrid intelligent algorithm to solve grain supply chain networ design problem. Differential Evolution (DE) algorithm is an evolutionary optimization method proposed by Storn and Price (995). Some researchers have used DE algorithm to solved manufacturing problems with mixed integer discrete variables (Lampinen and Zelina 999). In order to solve grain supply chain design problem, we present a hybrid intelligent algorithm to integrate 99-method (Liu 009) and differential evolution algorithm. 5

SIBO DING step. Initialize chromosomes whose feasibility is checed by the 99-method. There are 4 parameters. Population size NP is 00. Mutation factor F is 0.8. Crossover probability CR is 0.4. Max generation number is 000. The solution (chromosome) has 8 potential factories and 5 potential stores. Each factory has two genes: the first gene represents opening (=) or closing (=0) decisions. The second gene represents quantity of grain transported from a factory to stores. Each store has one gene representing an opening/closing decision to eep it open. step. Update the chromosomes by the mutation operation in which the 99-method is employed to chec the feasibility of offspring. step 3. Update the chromosomes by the crossover operation in which the 99-method is employed to chec the feasibility of offspring. step 4. Calculate the obective values for all chromosomes by the 99-method. step 5. Compute the fitness of each chromosome based on the obective value. step 6. Select the chromosomes by differential evolution algorithm selection operation. Integer values should be used to evaluate the obective function f(y i ), even though DE algorithm itself still wors internally with continuous floating-point values. step 7. Repeat the second to sixth steps a given number of maximum generation. step 8. Report the best chromosome as the optimal solution. Table : Uncertain normal distributions of grain quantity sold by farmers and potential locations of farmers No. 3 4 5 N(e i, σ i ) N(36, ) N(, 4) N(0, 3) N(0, ) N(38, 4) Coordinates(x,y) (5,3) (8,) (33,50) (37,55) (,45) No. 6 7 8 9 0 N(e i, σ i ) N(3, 3) N(35, 5) N(30, 4) N(5, 4) N(33, 4) Coordinates(x,y) (46,3) (34,5) (3,54) (3,48) (9,) No. 3 4 5 N(e i, σ i ) N(45, 3) N(8, ) N(40, ) N(46, 4) N(38, 4) Coordinates(x,y) (5,3) (8,) (33,50) (37,55) (,45) No. 6 7 8 9 0 N(e i, σ i ) N(, 4) N(, 5) N(3, 3) N(4, ) N(44, 3) Coordinates(x,y) (4,37) (,) (47,4) (0,4) (4,4) No. 3 4 5 N(e i, σ i ) N(39, ) N(6, ) N(5, 3) N(35, ) N(46, 3) Coordinates(x,y) (0,34) (33,) (8,4) (40,9) (8,3) Table : Uncertain normal distributions of setup costs, unit handling costs and potential locations of factories No. 3 4 N(e i, σ i ) N(000, 50) N(050, 5) N(950, 50) N(05, 5) h ($) 3.0.7 3.9 4. Coordinates(x,y) (9,44) (40,3) (4,8) (48,3) No. 5 6 7 8 N(e i, σ i ) N(00, 50) N(075, 5) N(095, 40) N(35, 50) h ($). 4.3 3.8 4.5 5.0 Coordinates(x,y) (7,34) (8,4) (33,46) (4,46) 6

A NEW UNCERTAIN PROGRAMMING MODEL FOR GRAIN SUPPLY CHAIN DESIGN Table 3: Uncertain normal distributions of setup costs and potential locations of stores No. 3 4 5 N(e i, σ i ) N(3000, 00) N(30, 50) N(390, 0) N(305, 40) N(330, 80) Coordinates(x,y) (,6) (6,4) (46,37) (37,) (6,30) 5 Numerical Example Suppose that the factories and stores are chosen from 8 and 5 potential sites to service 5 farmers. Euclidean distance is used for measuring travel distance, and the maximum allowable distance from farmers to the nearest factory is estimated to be 30 miles. Unit transportation cost c is.5$. Assume the cost confidence level α =0.8 and service confidence level β = 0.9, =,,, 8. ζ i, ξ and γ are assumed as independent normal uncertain variables. The next three tables give the value of the quantities. The best solution is 6.6546. Fig. shows a graphical representation of the best solution, where factory,, 3 are opened and accepted 63.344 units, 4.653 units and 47.8 units grain respectively, and stores,, 4 are opened. Figure : Setch map of grain supply chain 6 Conclusions The paper studied the grain supply chain design with uncertain quantity of grain and uncertain setup costs of factories and stores. In order to minimize the total cost, a new uncertain model for this problem was presented and was transformed into an equivalent deterministic programming problem. To solve the problem, a new hybrid intelligent algorithm that integrates 99-method and differential evolution algorithm was designed. Finally, a numerical example was presented in order to illustrate the effectiveness of the hybrid intelligent algorithm. Our wor offered an approach for constructing practical grain supply chain under uncertain environment. Acnowledgements This wor was supported by Soft Science Proect 0 of State Administration of Grain, High-level Talents Fund No.009BS03 of Henan University of Technology and Education Reform and Research Proect No.YPGC0-W03 and No.JZW0030. References [] Biswas, A., Pal, B.B., Application of Fuzzy Goal Programming Technique to Land Use Planning in Agricultural Systems, Omega, 33(005) 39-398. 7

SIBO DING [] Chen X.W., Ralescu Dan A., A Note on Truth Value in Uncertain Logic, Expert Systems with Applications, 38(0), 558-5586. [3] Darby-Dowman, K., Barer, S., Audsley, E., Parsons, D., A Two-stage Stochastic Programming Robust Planting Plans in Horticulture, Journal of the Operational Research Society, 5(000), 83-89. [4] Gao Y., Shortest Path Problem with Uncertain Arc Lengths, Computers and Mathematics with Applications, 6(0), 59-600. [5] Kaiser, H.M., Riha, S.J., Wils, D.S., Rossiter, D.G., Sampath, R., A Farm-level Analysis of Economic and Agronomic Impacts of Gradual Climate Warming, American Journal of Agricultural Economics,75(993), 387-398. [6] Lampinen, J., Zelina, I., Mechanical Engineering Design Optimization by Differential Evolution. In: Corne, D., Dorigo, M., Glover, F. (Eds.), New Ideas in Optimization. McGraw-Hill, London (UK), 999. [7] Liu, B., Uncertainty Theory, Springer-Verlag, Berlin, 007. [8] Liu B., Fuzzy Process, Hybrid Process and Uncertain Process, Journal of Uncertain Systems, (008), 3-6. [9] Liu, B., Theory and Practice of Uncertain Programming, nd ed.,springer-verlag, Berlin, 009. [0] Liu B., Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, 3(009), 3-0. [] Liu B., Uncertain Ris Analysis and Uncertain Reliability Analysis, Journal of Uncertain Systems, 4(00a), 63-70. [] Liu B., Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Control, Journal of Uncertain Systems, 4(00b), 83-98. [3] Liu, B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,Springer-Verlag, Berlin, 0. [4] Liu B., Uncertain Logic for Modeling Human Language, Journal of Uncertain Systems, 5(0a), 3-0. [5] Liu B., Why is there a need for uncertainty theory? Journal of Uncertain Systems, 6(0), 3-0. [6] Peng Z.X. and Iwamura Kauzo, A Sufficient and Necessary Condition of Uncertain Measure,Information: An International Interdisciplinary Journal, 5(0), 38-39. [7] Schilizzi S.G.M., Kingwell R.S., Effects of Climatic and Price Uncertainty on the Value of Legume Crops in a Mediterranean-type Environment,Agricultural Systems, 60(999), 55-69. [8] Storn R., Price K. Differential Evolution-A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces. ICSI, Technical Report TR-95-0,995. [9] Wang X.S., Gao Z.C., and Guo H.Y., Delphi Method for Estimating Uncertainty Distributions, Information: An International Interdisciplinary Journal, 5(0), 449-460. [0] Wei L., Xu J.P., Some Properties on Expected Value Operator for Uncertain Variables, Information: An International Interdisciplinary Journal, 3(00), 693-699. [] Yang X.H., Moments and Tails Inequality within the Framewor of Uncertainty Theory,Information: An International Interdisciplinary Journal, 4(0), 599-604. [] Zhou J., Liu B., Modeling capacitated location-allocation problem with fuzzy demands, Computers & Industrial Engineering, Vol.53, No.3, 454-468, 007. [3] Gao J., Liu B., Fuzzy multilevel programming with a hybrid intelligent algorithm, Computers & Mathematics with Applications, Vol.49, Nos.9-0, 539-548, 005. [4] Feng Y., Wu W., Zhang B., and Gao J., Transmission line maintenance scheduling considering both randomness and fuzziness, Journal of Uncertain Systems, Vol.5, No.4, 43-56, 0. [5] Gao J., Zhang Q., Shen P., Coalitional game with fuzzy payoffs and credibilistic Shapley value, Iranian Journal of Fuzzy Systems, Vol.8, No.4, 07-7, 0. [6] Shen P., Gao J., Coalitional game with fuzzy payoffs and credibilistic core, Soft Computing, Vol. 5, No.4, 78-786, 0. [7] Liu Y.K., Liu B., Random fuzzy programming with chance measures defined by fuzzy integrals, Mathematical and Computer Modelling, Vol.36, No.4-5, 509-54, 00. 8