Abstract
Fuzzy Inf. Eng. (2011) 2: 193-208 DOI 10.1007/s12543-011-0077-6 ORIGINAL ARTICLE A Linear Programming Priority Method for a Fuzzy Transportation Problem with Non-linear Constraints — The Case of a General Contractor Company Hossein Abdollahnejad Barough Received: 30 January 2011/ Revised: 25 April 2011/ Accepted: 18 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract Demand and supply pattern for most products varies during their life cy- cle in the markets. In this paper, the author presents a transportation problem with non-linear constraints in which supply and demand are symmetric trapezoidal fuzzy value. In order to reﬂect a more realistic pattern, the unit of transportation cost is assumed to be stochastic. Then, the non-linear constraints are linearized by adding auxiliary constraints. Finally, the optimal solution of the problem is found by solving the linear programming problem with fuzzy and crisp constraints and by applying fuzzy programming technique. A new method proposed to solve this problem, and is illustrated through numerical examples. Multi-objective goal programming method- ology is applied to solve this problem. The results of this research were developed and used as one of the Decision Support System models in the Logistics Department of Kayson Co. Keywords Fuzzy transportation · Non-linear programming · Multi-objective pro- gramming· Fuzzy linear programming· Stochastic programming 1. Introduction In general, distribution of products from sources to destinations is called “Transporta- tion Problem”, which was ﬁrst developed by F. L. Hitchcock in 1941. It usually aims to minimize the total transportation cost. Other objectives may include minimization of delivery time, maximization of the proﬁt, etc [17]. The classical transportation problem refers to a special class of linear programming problems. In a typical trans- portation problem, a product is to be transported from M sources to N destinations Hossein Abdollahnjead Barough () Department of Industrial Engineering, Payame Noor University, Tehran, Iran. email: hossein.abdolahnejad@gmail.com 194 Hossein Abdollahnejad Barough (2011) and their demand and supply values are a , a , ··· , a and b , b ,··· , b respec- 1 2 M 1 2 N tively. In addition, there is a penalty c associated with transporting a unit of the ij product from source i to destination j. This penalty may be cost or delivery time or safety of delivery, etc. In this paper, two goal programming models are reformulated in order to make sup- ply and demand fuzzy values, where the multiple aspiration levels of the cost goal are handled by the use of multiplicative terms of the binary variables. The organization of this paper is as follows: Section 2 reviews some basic deﬁnitions and assumptions of the area of fuzzy theory. Section 3 deals with previous works on fuzzy transportation models. Two transportation models and their respective goal programming formula- tions are presented in Section 4. Section 5 explains the method for solution of the linear programming problem with fuzzy and crisp constraints. Two numerical exam- ples of the proposed method are presented in Section 6. These numerical examples are modiﬁed real problems faced by the Logistics Department of Kayson Company. Finally, Section 7 gives some concluding remarks on the proposed method. 2. Preliminaries We review the fundamental notions of fuzzy set theory (taken from [13,15]). Deﬁnition 1 A fuzzy number a ˜ on R is said to be a symmetric trapezoidal fuzzy num- L U L U ber if there exist real numbers a and a ,a ≤ a andα ≥ 0 such that ⎪ x α− a L L ⎪ + , x ∈ [a −α, a ], ⎪ α α ⎪ L U 1, x ∈ [a , a ], a ˜ (x) = ⎪ −x a +α U U ⎪ + , x ∈ [a , a +α], α α 0, otherwise. L U L We denote symmetric trapezoidal fuzzy number a ˜ by a ˜ = (a , a ,α), where (a − U L U α, a + α) is the support of a ˜ and [a , a ] its core, and the set of all symmetric trapezoidal fuzzy numbers by F(R). L U L U Let a ˜ = (a , a ,α) and b = (b , b ,β) be two symmetric trapezoidal fuzzy num- bers. Then the arithmetic operations on a ˜ and b are given as follows: L L U U Addition: a ˜ + b = (a + b , a + b ,α+β). L U U L − b , a − b ,α+β). Subtraction: a ˜ − b = (a L U L U L U L U a + a b + b a + a b + b U U Multiplication: a ˜ b = (( )( )−ω, ( )( )+ω,|a β+b α|), 2 2 2 2 where t − t 2 1 ω = , L L L U U L U U L L L U U L U U t = min{a b , a b , a b , a b }, t = max{a b , a b , a b , a b }. 1 2 From the above deﬁnition it is clear that L U λ> 0,λ ∈ R; λ a ˜ = (λa ,λa ,λα), U L λ< 0,λ ∈ R; λ a ˜ = (λa ,λa ,−λα). Fuzzy Inf. Eng. (2011) 2: 193-208 195 Note that depending upon the need, we can also use a smaller ω in the deﬁnition of multiplication involving symmetric trapezoidal fuzzy numbers. L U L U Deﬁnition 2 Let a ˜ = (a , a ,α) and b = (b , b ,β) be two symmetric trapezoidal fuzzy numbers. Deﬁne the relation as L U L U (a −α)+ (a +α) (b −β)+ (b +β) ˜ ˜ a ˜ b (or b a ˜ ) if and only if either < , 2 2 L U L U a + a b + b that is < (in this case, we may write a ˜ ≺ b), 2 2 L U L U a + a b + b L L U U or = , b < a and a < b , 2 2 L U L U a + a b + b L L U U or = , b = a , a = b andα ≤ β. 2 2 ˜ ˜ Note that in the two above cases, we also write a ˜ ≈ b and say that a ˜ and b are equivalent. L U L U Remark 1 Two symmetric trapezoidal fuzzy numbers a ˜ = (a , a ,α), b = (b , b ,β) L U L U a + a b + b are equivalent if and only if = . 2 2 Deﬁnition 3 For any trapezoidal fuzzy number a, ˜ we deﬁne a ˜ 0, if there existε ≥ 0 ˜ ˜ and α ≥ 0 such that a ˜ (−ε,ε,α). We also denote (−ε,ε,α) by 0. Note that 0 is equivalent to (0, 0, 0) = 0. Naturally, one may consider 0 = (0, 0, 0) as zero symmetric triangular fuzzy number. Remark 2 If x ˜ ≈ 0, then x ˜ is said to be a zero symmetric trapezoidal fuzzy number. ˜ ˜ ˜ It is to be noted that if x ˜ = 0, then x ˜ ≈ 0, but the converse need not be true. If x ˜ 0 (that is x ˜ is not equivalent to 0), then is said to be a non-zero symmetric trapezoidal ˜ ˜ fuzzy number. It is to be noted that if x ˜ 0, then x ˜ 0, but the converse need not ˜ ˜ ˜ be true. If x ˜ 0(x˜ 0) and x ˜ 0, then is said to be a positive (negative) symmetric ˜ ˜ trapezoidal fuzzy number and is denoted by x ˜ 0(x˜ ≺ 0). ˜ ˜ ˜ ˜ Now if a ˜ , b ∈ F(R), it is easy to show that if a ˜ b, then a ˜ − b 0. The following lemma immediately follows form Deﬁnition 1. Lemma 1 If a ˜, b ∈ F(R), and c ∈ R such that c 0, then ˜ ˜ (i) a ˜ b ≈ b a. ˜ ˜ ˜ ˜ (ii) c(˜ a b) ≈ (ca ˜ ) b ≈ a ˜ (cb). Two following results taken from [14] and we omit the details and proofs here. Lemma 2 For any symmetric trapezoidal fuzzy number a ˜, b and c, ˜ we have ˜ ˜ (i) c ˜(˜ a+ b) ≈ (˜ ca ˜ + c ˜b). ˜ ˜ (ii) c ˜(˜ a− b) ≈ (˜ ca ˜ − c ˜b). Here we also give some new results on arithmetic of the fuzzy numbers. 196 Hossein Abdollahnejad Barough (2011) Lemma 3 If a ˜, b ∈ F(R), then ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ (i) a ˜ b 0, if and only if a ˜ 0 and b 0 or a ˜ 0 and b 0. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ (ii) a ˜ b 0, if and only if a ˜ 0 and b 0,or a ˜ 0 and b 0. ˜ ˜ (iii) a ˜ b, if and only ifλa ˜ λb, for anyλ< 0,λ ∈ R. Proof It is straightforward. ˜ ˜ Lemma 4 Suppose that a ˜, b, c ˜ ∈ F(R) such that a ˜ b, then ˜ ˜ (i) c ˜a ˜ c ˜b, if c ˜ 0. ˜ ˜ (ii) c ˜a ˜ c ˜b, if c ˜ 0. ˜ ˜ ˜ ˜ ˜ Proof From a ˜ b,wehave b− a ˜ 0. Hence from Lemma 3, we have c ˜(b− a ˜ ) 0, ˜ ˜ ˜ ˜ if c ˜ 0, and also c ˜(b− a ˜ ) 0, if c ˜ 0. Now the results are clear by Lemma 2. ˜ ˜ ˜ ˜ Lemma 5 Suppose that a ˜, b, c ˜ ∈ F(R) such that a ˜ ≈ b, c ˜ 0, then, c ˜a ˜ ≈ c ˜b. Proof It is straightforward. 3. Literature Review 3.1. Linguistic Variables and Fuzzy Sets A linguistic variable is a variable whose values are words or sentences in a natural or artiﬁcial language [21]. For instance, age is a linguistic variable if its values are assumed to be the fuzzy variables labelled ‘not young’, ‘young’ and ‘very young’ rather than the actual numbers. The concept of a linguistic variable provides a means for the approximate characterization of phenomena that are too complex or too ill- deﬁned to be amenable to description in conventional quantitative terms. The main applications of the linguistic approach lay in the realm of humanistic systems, es- pecially in the ﬁelds of artiﬁcial intelligence, linguistics, human decision processes, pattern recognition, psychology, law, medical diagnosis, information retrieval, eco- nomics and related areas [21]. 3.2. Fuzzy Linear Programming First introduced by Zadeh [23], the concept of the fuzzy set theory is used for solv- ing different types of linear programming (LP) problems. Zimmermann [24] ﬁrst introduced fuzzy linear programming (FLP) as conventional LP. He used linear mem- bership functions and the minimizing operator as an aggregator for these functions, and assigned an equivalent LP to fuzzy linear programming problems. Subsequently, Zimmermann’s fuzzy linear programming has developed into several fuzzy optimiza- tion methods for solving transportation problems. He presented a fuzzy approach to multi-objective linear programming problems [24]. He also studied the duality re- lations in fuzzy linear programming [24]. Fuzzy linear programming problem with fuzzy coefﬁcients was formulated by Negoita [18] and called robust programming. Dubois and Prade [14] investigated linear fuzzy constraints. Tanaka and Asai [18] Fuzzy Inf. Eng. (2011) 2: 193-208 197 also proposed a formulation of fuzzy linear programming with fuzzy constraints and devised a method based on inequality relations between fuzzy numbers. 3.3. Fuzzy Transportation Problems In practice, the parameters of transportation problem, i.e., demand and supply val- ues are not always exactly known and stable. Chanas and Kuchta [6] proposed the concept of the optimal solution for the transportation problem with fuzzy cost coef- ﬁcients expressed as L-R fuzzy numbers, and developed an algorithm for obtaining the optimal solution. Additionally, Chanas and Kuchta [7] designed an algorithm for solving integer fuzzy transportation problems with fuzzy demand and supply values with a view to maximize the joint satisfaction of the fuzzy goal and the constraints. 3.4. Goal Programming The term ‘Goal Programming’ was introduced by Charnes and Cooper [12] in 1961. Decision makers sometimes set such goals, even when they are unattainable within the available resources. Such problems are tackled with the help of the techniques of goal programming. Any constraint incorporated is called a goal. Whether the goals are attainable or not, the objective function is stated in such a way that it’s op- timization means as ‘close as possible’ to the indicated goals. Multi-objective linear programming problems exist in many managerial decision making problems. Hiller and Lieberman [16] and Ravindran et al [20] have considered a mathematical model in which an appropriate constraint is to be chosen using binary variables. A method for modelling the multi-objective goal programming problem, using the multiplica- tive terms of binary variables to handle the multiple aspiration levels was presented by Chang [9]. He has also given a method where the multiplicative terms of the binary variables are replaced by a continuous variable [11]. 3.5. Multi-objective Goal Programming Transportation Problems The multi-objective transportation model is set to solve the transportation problem si- multaneously associated with several objectives. Normally, existing multi-objective transportation models use a minimization of the total cost objective as one of their ob- jectives. The other objectives may include the quantity of goods delivered, underused capacity, energy consumption, total delivery time, etc [20]. 4. Mathematical Models 4.1. The Transportation Model Network The transportation problem (TP) was ﬁrst developed and proposed by F. L. Hitchcock in 1941 [19]. The classical transportation problem is referred to a special case of LP problem and its model is applied to determine an optimal solution to delivery avail- able amount of satisﬁed demand in which the total transportation cost is minimized [19]. The transportation problem network form can be shown as in Figure 1. 4.2. The Transportation Model Network The following notations are applied to describe the transportation problem. Index Sets: 198 Hossein Abdollahnejad Barough (2011) M Number of sources (i = 1, 2,··· , m); N Number of destinations (j = 1, 2,··· , n); th A Fuzzy quantity of i source; th B Fuzzy quantity of j destination; th th C Stochastic unit of transportation cost from i source to j , ij destination; + + + th p , e , p Over achievements of i goal; − − − th p , e , p Under achievements of i goal; λ Maximum degree of satisfaction of the fuzzy constraints; a,b Crisp numbers; th η Penalty of each unit of over/under achievements of i goal; f Objective function; f Optimal value of lower boundλ - cut; f Optimal value of upper boundλ - cut; μ Membership function. Decision Variables: th th X The amount of demand transported from i to j ij destination. Fig.1 The transportation problem network 4.3. Transportation Linear Programming Problem Consider the transportation model: Fuzzy Inf. Eng. (2011) 2: 193-208 199 M N min f (x) = C = (C , C ,··· , C ) = C(x ) ij 1 2 k ij i=1 j=1 Subject to 1 2 3 4 (1) x (a , a , a , a ), (i = 1, 2,··· , M), ij i i i i j=1 1 2 3 4 x (b , b , b , b ), ( j = 1, 2,··· , N), ij j j j j i=1 x ≥ 0 and are integers. ij The objective function can assume only one of k choices (aspiration levels) b , b , ··· , b . We illustrate the procedure for ﬁnding an optimal solution of the above 2 k problem for the case with four goals. Since the number of choices is 4 (= 2 )two binary variables are required to model the situation. We rewrite (1) equations for four choices case as follows: C(x) = z z c + (1− z )z c + z (1− z )c + (1− z )(1− z )c = ϕ 1 2 1 1 2 2 1 2 3 1 2 4 Subject to 1 2 3 4 x (a , a , a , a ), (i = 1, 2,··· , M), ij i i i i j=1 (2) 1 2 3 4 x (b , b , b , b ), ( j = 1, 2,··· , N), ij j j j i=1 x ≥ 0 and are integers, ij z , z are 0or1. 1 2 4.4. Multi-Objective Transportation Problem (MOTP) In order to minimize ϕ, a ﬂexible membership function goal with the aspired level 1(i.e., the highest possible value of membership function) is used as follows, (Goal −ϕ) max + − (3) − p + p = 1, i i (Goal − Goal ) max min where Goal and Goal are respectively upper and lower bounds of the aspiration max min + − levels of the cost goal, and p and p are respectively over and under achievements of i i th the i goal. Using the goal programming method presented by Chang [9] for a linear programming problem with minimization type objective function, we construct the following goal programming problem for Problem (2): 200 Hossein Abdollahnejad Barough (2011) + − + − min p + p + p + p 1 1 2 2 Subject to 1 2 3 4 x (a , a , a , a ), (i = 1, 2,··· , M), ij i i i i j=1 1 2 3 4 x (b , b , b , b ), ( j = 1, 2,··· , N), ij j j j j i=1 M N (4) + − c x − p + p = ϕ, ij ij 1 1 i=1 j=1 ϕ = z z c + (1− z )z c + z (1− z )c + (1− z )(1− z )c , 1 2 1 1 2 2 1 2 3 1 2 4 ϕ Goal min + − − p + p = , 2 2 (Goal − Goal ) (Goal − Goal ) max min max min x ≥ 0 and are integers, ij z , z are0or1. 1 2 4.5. Non-linear Multi-objective Transportation Problem (NLMOTP) Lineariza- tion The Non-linear constraints of the above problem can be linearized by deﬁning (z = z z ) and adding the linear constraint (z + z − 1 ≤ 2z ≤ z + z ) in which z =0or 1 2 1 2 3 1 2 3 1. Hence (4) can be written as: + − + − min p + p + p + p 1 1 2 2 Subject to 1 2 3 4 x (a , a , a , a ), (i = 1, 2,··· , M), ij i i i i j=1 1 2 3 4 x (b , b , b , b ), ( j = 1, 2,··· , N), ij j j j i=1 M N (5) + − c x − p + p = ϕ, ij ij 1 1 i=1 j=1 ϕ = (c − c )z + (c − c )z + (c − c − c − c )z + c , 2 4 1 3 4 2 1 2 3 4 3 4 ϕ Goal min + − − p + p = , 2 2 (Goal − Goal ) (Goal − Goal ) max min max min x ≥ 0 and are integers, ij z , z , z are0or1. 1 2 3 4.6. Strategy for Solving the Imprecise Objective Function Consider the following transportation model where the cost goal can assume any value in a prescribed range: Fuzzy Inf. Eng. (2011) 2: 193-208 201 a ≤ y = c(x) ≤ b Subject to 1 2 3 4 x (a , a , a , a ), (i = 1, 2,··· , M), ij i i i i (6) j=1 1 2 3 4 x (b , b , b , b ), ( j = 1, 2,··· , N), ij j j j j i=1 x ≥ 0 and are integers. ij A penalty η is assigned for exceeding the cost goal and there is no penalty for achieving a value lesser than the aspiration levels. Using the goal programming method given by Chang[11] for a linear programming problem with minimization type objective function, we construct the following goal programming problem: + + − min η(p )+ e + e Subject to M N + − c x − p + p = y, ij ij i=1 j=1 (7) + − y− e + e = a, a ≤ y ≤ b, + − + − p , p , e , e ≥ 0, x ≥ 0 and are integers. ij 5. Solution Presentation Problems (5) and (6) are linear programming problems with fuzzy and crisp con- 1 2 3 4 straints as where A =[b , b , b , b ] is a column vector of trapezoidal fuzzy numbers. j j j j This problem can be solved by generalizing the method. The membership function of the objective function of Problem (7) can be determined by solving the following two linear programming problems: min f (x ) ij Subject to 4 4 3 ∗ A x ≤ a − (a − a )λ , i ij i i i (8) 1 2 1 ∗ A x ≥ a − (a − a )λ , i ij i i i A x ≤ B , i ij j x ≥ 0 ij yielding the optimal value f and min f (x ) ij Subject to A x ≤ a , i ij (9) A x ≥ a , i ij A x ≤ B , i ij j x ,≥ 0 ij 202 Hossein Abdollahnejad Barough (2011) yielding the optimal value f . The membership function of the objective function of Problem (7) is: 1, if f (x ) ≤ f , ⎪ ij 0 ⎨ ( f (x j)− f ) i 1 μ (x ) = (10) ˜ ⎪ , if f ≤ f (x ) ≤ f , ij 0 ij 1 f ⎪ ( f − f ) 0 1 0, if f (x ) ≥ f ij 1 by applying fuzzy programming technique, we get: max λ Subject to ( f − f )λ+ f (x ) ≥ f , 0 1 ij 1 4 3 4 (a − a )λ+ A x ≤ a , (11) i ij i i i 2 1 1 −(a − a )λ+ A x ≥ a , i ij i i i A x ≤ B , i ij j x ≥ 0. ij The optimal solution of the above Linear Programming gives the optimal solution to the considered fuzzy transportation problem. 6. Numerical Examples 6.1. Case with Discrete Objectives c(x ) = 2x + 3x + 4x + 2x [9, 10, 11, 12] ij 11 12 21 22 Subject to x + x [1, 2, 3, 4], 11 12 (12) x + x [2, 3, 5, 6], 21 22 x + x [1, 2, 3, 5], 11 21 x + x [2, 4, 5, 6], 12 22 x ≥ 0for iandj=1,2andare integers. ij The goal programming formulation using (4) is: + − + − min p + p + p + p 1 1 2 2 Subject to + − c(x) = 2x + 3x + 4x + 2x − p + p = ϕ, 11 12 21 22 1 1 x + x [1, 2, 3, 4], 11 12 x + x [2, 3, 5, 6], 21 22 (13) x + x [1, 2, 3, 5], 11 21 x + x [2, 4, 5, 6], 12 22 ϕ+ 2z + z = 12, 1 2 + − ϕ+ 3p − 3p = 9, 2 2 x ≥ 0for iandj=1,2andare integers. ij We solve the following two linear programming problems from (8) and (9): Fuzzy Inf. Eng. (2011) 2: 193-208 203 + − + − min p + p + p + p 1 1 2 2 Subject to + − c(x) = 2x + 3x + 4x + 2x − p + p −ϕ = 0, 11 12 21 22 1 1 x + x ≤ 3, 11 12 x + x ≥ 2, 11 12 x + x ≤ 5, 21 22 x + x ≥ 3, 21 22 (14) x + x ≤ 3, 11 21 x + x ≥ 2, 11 21 x + x ≤ 5, 12 22 x + x ≥ 4, 12 22 ϕ+ 2z + z = 12, 1 2 + − ϕ+ 3p − 3p = 9, 2 2 x ≥ 0for iandj=1,2andare integers. ij The optimal solution of this linear programming problem is x = 2, x = 4, 11 22 p = 4, ϕ = 12 with the optimal value of the objective function f = 1: + − + − min p + p + p + p 1 1 2 2 Subject to + − c(x) = 2x + 3x + 4x + 2x − p + p −ϕ = 0, 11 12 21 22 1 1 x + x ≤ 4, 11 12 x + x ≥ 1, 11 12 x + x ≤ 6, 21 22 x + x ≥ 2, 21 22 x + x ≤ 5, 11 21 (15) x + x ≥ 1, 11 21 x + x ≤ 6, 12 22 x + x ≥ 2, 12 22 ϕ+ 2z + z = 12, 1 2 + − ϕ+ 3p − 3p = 9, 2 2 x ≥ 0for iandj=1,2andare integers, ij + − + − ϕ, p , p , p , p ≥ 0, 1 1 2 2 z and z = 0or1. 1 2 The optimal solution of this linear programming problem is x = 1, x = 1, 12 21 x = 1, p = 4, ϕ = 12 with the optimal value of the objective function f = 0. 22 0 Hence from (11) we have : 204 Hossein Abdollahnejad Barough (2011) min λ Subject to + − + − λ+ p + p + p + p ≥ 1, 1 1 2 2 x + x +λ ≤ 4, 11 12 x + x −λ ≥ 1, 11 12 x + x +λ ≤ 6, 11 22 x + x −λ ≥ 2, 21 22 x + x +λ ≤ 5, 11 21 (16) x + x −λ ≥ 1, 11 21 x + x +λ ≤ 6, 12 22 x + x −λ ≥ 2, 12 22 ϕ+ 2z + z = 12, 1 2 + − ϕ+ 3p − 3p = 9, 2 2 x ≥ 0foriandj=1,2andare integers, ij + − + − ϕ, p , p , p , p ≥ 0, 1 1 2 2 z and z = 0or1. 1 2 Hence the optimal solution of Problem (12) is x = 2, x = 4, λ = 1, ϕ = 12. 11 22 6.2. Case with Continuous Objectives 8 ≤ c(x) = 2x + 3x + 4x + 2x ≤ 10 11 12 21 22 Subject to x + x [1, 2, 3, 4], 11 12 (17) x + x [2, 3, 5, 6], 21 22 x + x [1, 2, 3, 5], 11 21 x + x [2, 4, 5, 6], 12 22 x ≥ 0 i and j =1, 2 and are integers. ij Penalty 2 is assigned for exceeding the cost goal. The goal programming formula- tion is: + + − min 2p + e + e Subject to x + x [1, 2, 3, 4], 11 12 x + x [2, 3, 5, 6], 21 22 x + x [1, 2, 3, 5], 11 21 x + x [2, 4, 5, 6], (18) 12 22 + − 2x + 3x + 4x + 2x − p + p − y = 0, 11 12 21 22 + − y− e + e = 8, 8 ≤ y ≤ 10, x ≥ 0 i and j =1, 2 and are integers, ij + − + − p , p , e , e ≥ 0. Fuzzy Inf. Eng. (2011) 2: 193-208 205 We solve the following two linear programming problems from (8) and (9): + + − min 2p + e + e Subject to x + x ≤ 3, 11 12 x + x ≥ 2, 11 12 x + x ≤ 5, 21 22 x + x ≥ 3, 21 22 x + x ≤ 3, 11 21 (19) x + x ≥ 2, 11 21 x + x ≤ 5, 12 22 x + x ≥ 4, 12 22 + − 2x + 3x + 4x + 2x − p + p − y = 0, 11 12 21 22 + − y− e + e = 8, 8 ≤ y ≤ 10, x ≥ 0 i and j =1, 2 and are integers, ij + − + − p , p , e , e ≥ 0. The optimal value of the object function is f : + + − min 2p + e + e Subject to x + x ≤ 4, 11 12 x + x ≥ 1, 11 12 x + x ≤ 6, 21 22 x + x ≥ 2, 21 22 x + x ≤ 5, 11 21 (20) x + x ≥ 1, 11 21 x + x ≤ 6, 12 22 x + x ≥ 2, 12 22 + − 2x + 3x + 4x + 2x − p + p − y = 0, 11 12 21 22 + − y− e + e = 8, 8 ≤ y ≤ 10, x ≥ 0 i and j =1, 2 and are integers, ij + − + − p , p , e , e ≥ 0. The optimal value of the objective function is f = 0 . Hence from (11) we have, : 0 206 Hossein Abdollahnejad Barough (2011) max λ Subject to x + x +λ ≤ 4, 11 12 x + x −λ ≥ 1, 11 12 x + x +λ ≤ 6, 21 22 x + x −λ ≥ 2, 21 22 x + x +λ ≤ 5, 11 21 (21) x + x −λ ≥ 1, 11 21 x + x +λ ≤ 6, 12 22 x + x −λ ≥ 2, 12 22 + − 2x + 3x + 4x + 2x − p + p − y = 0, 11 12 21 22 + − y− e + e = 8, 8 ≤ y ≤ 10, x ≥ 0 i and j =1, 2 and are integers, ij + − + − p , p , e , e ≥ 0. The optimal solution of Problem (17) is x = 2, x = 4. ϕ = 12, λ = 1. 11 22 7. Conclusion Transportation models have wide applications in Logistics and Supply Chain Man- agement and reduction of costs. In real world applications, the parameters in trans- portation problems may not been known precisely due to uncontrollable factors. If the obtained results are crisp values it might lose some helpful information. Since the objective value is expressed by the membership function rather than by a crisp value, more information is provided for making decisions. Some previous studies have devised solution procedures for fuzzy transportation problems. The objective value derived from those studies is crisp values rather than fuzzy numbers. This study presents an MOTP with stochastic unit of transportation cost and symmetric trapezoidal fuzzy demand and supply values. In this study, the traditional transporta- tion problem is reconstructed with a multi-objective goal programming approach, then the problem is linearized by deﬁning auxiliary constraints. Finally the optimal solution of the model developed with fuzzy and crisp constraints is founded through applying fuzzy programming technique by the deﬁned fuzzy membership function. Two numerical examples demonstrate the feasibility of applying the multi-objective goal programming approach to fuzzy transportation problems. These examples are the real modiﬁed cases in Logistics Department of Kayson Co. We hope this study helps to stimulate future work on the multi-objective goal pro- gramming area. Future work may include applications to some practical problems like machine scheduling, transportation, logistics and location allocation. Our general results, when applied to these special problems, may trigger the development of some specialized procedures. Moreover, some specialized optimization or approximation algorithms may be designed for some particular constrained optimization problems and this would reduce the time to reach the exact or approximate non-dominated so- lutions. Fuzzy Inf. Eng. (2011) 2: 193-208 207 However, there are other key areas which need to be studied and researched further including: 1. The model needed to study more for solving fuzzy transportation programming problems with non-basic optimal solution. 2. The performance of the fuzzy approach needs to be evaluated and examined more with various numerical simulations. 3. A careful comparison study must be carried out between the fuzzy and crisp approaches to transportation problems. 4. Promising research area may include the development of the procedures for the optimal solution of the pre-speciﬁed function of the n-objectives. Acknowledgements The author respectively thanks to Prof. B. Y. Cao, the Editor-in-Chief of the Fuzzy Information and Engineering—an international journal, Dr. S. H. Nasseri, the As- sistant Professor of the Departments of Mathematics and Computer Sciences of the Mazandaran University and the Senior Lecturer of the Guangzhou University for his kindly supports. Also to Mr. M. T. Azin, the Public Relations Manager of Kayson Co. for his supports and editing english errors on early drafts of this paper and anony- mous referees for their helpful comments on this paper. All errors are those of the author. References 1. Bellman R R, Zadeh L A (1970) Decision making in a fuzzy environment. Management Science 17: 203-218 2. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2011
Keywords: Fuzzy transportation; Non-linear programming; Multi-objective programming; Fuzzy linear programming; Stochastic programming