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A class of random fuzzy programming and its application to supply chain design
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Computers & Industrial Engineering 56 (2009) 937–950 www.elsevier.com/locate/caie
A class of random fuzzy programming and its application to supply chain design Jiuping Xu a,*, Yanan He b, Mistsuo Gen c a
Uncertainty Decision-Making Laboratory, School of Business and Administration, Sichuan University, Chengdu 610064, China b Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China c Graduate School of Information, Production and Systems, Waseda University, Kitakyushu 808-0315, Japan Available online 14 October 2008
Abstract Supply chain design problems have recently raised a lot of interest since the opportunity of an integrated management of the supply chain can reduce the propagation of undesirable events through the network and can affect decisively the profitability of the members. Often uncertainties may be associated with demand, supply or various relevant costs. In most of the existing models uncertainties are treated as randomness and are handled by appealing to probability theory. However, in certain situations uncertainties are due to random fuzziness and in these case the random fuzzy programming is applicable, while no attempt has been made to define those uncertain parameters as random fuzzy variables. We propose a random fuzzy programming model and a methodology for solving a multi-stage supply chain design problem of a realistic scale in the random fuzzy environment. Based on the concept of random fuzzy variable, we use probability theory, fuzzy set theory and optimizing theory as our research tool. At a given possibility a, we provide an auxiliary programming model of the random fuzzy programming and convert it to a deterministic mixed 0–1 integer programming model, of which the optimal solutions are proved to exist. Then we present a novel technique called spanning-tree based on genetic algorithm (st-GA) to get the heuristic solutions. A computational example involving a relatively large supply chain is presented to highlight the significance of the random fuzzy model as well as the effectiveness of the proposed solution method. Random fuzzy variable is a very useful tool to describe the uncertain parameters and our methodology may be applied to any optimization problems with random fuzzy factors. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Random fuzzy variable; Random fuzzy programming; Supply chain design; st-GA
1. Introduction A supply chain is a network comprised of a set of geographically dispersed facilities (suppliers, plants, and warehouses or distribution centers) where raw materials, intermediate products, or finished products are acquired, transformed, stored, or sold. It is often regarded as the art of bringing the right amount of the right
*
Corresponding author. Tel.: +86 28 85418522; fax: +86 28 85400222. E-mail addresses: [email protected], [email protected] (J. Xu), [email protected] (Y. He).
0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.09.045
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product to the right place at the right time. If the facilities are to distribute product directly to customers, then single-stage model is appropriate. On the other hand, if several facilities are to be sited between the suppliers to the customers in order to produce product or act as regional warehouses or distribution centers, then multistage model is the appropriate model. The structure of the aforementioned multi-stage supply chain is depicted in Fig. 1. Mathematical programming models have proven their usefulness as analytical tools to optimize complex decision-making problems such as those encountered in supply chain planning. Geoffrion and Graves (1974) described a multi-commodity distribution system design problem and solved it by Benders Decomposition. This is probably the first paper that presents a comprehensive MIP model for the strategic design of supply chain networks. After that, a diversity of deterministic mathematical programming models dealing with the design of supply chain networks can be found in the literature. See for example Aikens (1985), Goetschalckx, Vidal, and Dogan (2002), Geoffrion and Powers (1995), Yan, Yu, and Cheng (2003), and Amiri (2004). Under the most circumstances, the critical design parameters for the supply chain, such as customer demands, prices and resource capacities are generally uncertain. Moreover, the arrival of regional economic alliances, for instance the Asian Pacific Economic Alliance and the European Union, have prompted many corporations to move more and more towards global supply chains, and therefore to become exposed to risky factors such as exchange rates, reliability of transportation channels, and transfer prices (Santoso, Ahmed, Goetschalckx, & Shapiro, 2005). Unless the supply chain is designed to be robust with respect to the uncertain operating conditions, the impact of operational inefficiencies such as delays and disruptions will be larger than necessary. Uncertain supply chain design has been one of the promising subjects. Beginning with the seminal works of Charnes and Cooper (1959), optimization under uncertainty has experienced rapid development in both theory and algorithms. A big amount of stochastic programming models have been proposed for strategic and tactical planning. See for example Owen and Daskin (1998), Cheung and Powell (1996), Guille´n, Mele, Bagajewicz, Espun˜a, and Puigjaner (2005), Min and Zhou (2002), and Landeghen and Vanmaele (2002). However, in certain situations, the assumption of precise parameters of probability distributions is seriously questioned. The parameters are fixed statistically estimated using past demand information, while demand does not stay ‘static’ in fact. When the conditioning variables, such as the technologic innovations and preferences of consumers, considerably change, the mean and variance of the demand distribution are possible to change. Besides, it is almost impossible to specify exactly the true values to the parameters, especially in the absence of abundant information as in the case of demand of new products. It is more appropriate to estimate the parameters ‘‘approximately a”. Thus, based on expert experience fuzzy variables are considered to describe them. In this case, random variables with imprecise parameters are random fuzzy variables. By random fuzzy programming we mean the optimization theory in random fuzzy environment. Random fuzzy programming makes the supply chain design plan more flexible when the parameters of the coefficients’
Fig. 1. Supply chain structure.
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distribution are uncertain. In practice, the values of the fuzzy parameters can be obtained according to the expert experience. Different numbers of the fuzzy variables reflect different conditions which affect the parameters of the probability distribution. The concept of random fuzzy variables was provided by Liu (2002), which is different from the definition used by Nattier (2001). To the best of our knowledge, there is a little research for the programming and solving supply chain design problems in random fuzzy environment. Therefore, there is strong motivation for further research in the area. We wish to highlight the significance of introducing random fuzziness in supply chain design modeling. In this paper, we consider a multi-stage supply chain design problem with a number of potential facilities (plants, warehouses) and the task includes the choices of facilities to be opened and the distribution pattern design to satisfy the demand with minimum cost in random fuzzy environment. A random fuzzy programming model is proposed and a methodology is developed. The remainder of this paper is organized as follows: in the next section, we state some basic concepts and results on random fuzzy variables and optimizing theory. In Section 3, we develop a random fuzzy mathematical programming model for a multi-stage supply chain designing problem in a random fuzzy environment. In Section 4, we present our methodology, and discuss the existence of solution. In Section 5, we introduce the stGA to get the heuristic solutions. Section 6 provide a real-life multi-stage supply chain design test case. Finally we offer some concluding remarks in Section 7. 2. Preliminaries In this section, we will state some basic concepts and theorems on random fuzzy variables and optimizing theory. These results are crucial for the remainder of this paper. Interested readers may consult Liu (2002) where important properties of random fuzzy variables are recorded. Definition 2.1. (Liu, 2002) A random fuzzy variable is a function from the possibility space (H; PðHÞ; Pos) to a collection of random variables R, where H is a universe, PðHÞ is the power of H, and Pos is a possibility measure defined on PðHÞ. Definition 2.2. (Liu, 2002) Let n be a random fuzzy variable on the possibility space (H; PðHÞ; Pos). Then its membership function is derived from the possibility measure Pos by lðgÞ ¼ Posfh 2 HjnðhÞ ¼ gg;
g2R
ð2:1Þ
The a-cut na is defined as na ¼ fg 2 Rjln ðgÞ P ag
ð2:2Þ
Let an n-dimensional random fuzzy vector n ¼ ðn1 ; n2 ; . . . ; nn Þ. Then we can introduce the following a-level set or a-cut of n. Definition 2.3. The a-level (a 2 ½0; 1) set of the random fuzzy vector n ¼ ðn1 ; n2 ; . . . ; nn Þ is defined as the ordinary set La ðnÞ: La ðnÞ ¼ fg ¼ ðg1 ; g2 ; . . . ; gn Þjln1 ðg1 Þ P a; ln2 ðg2 Þ P a; . . . ; lnn ðgn Þ P ag ð2:3Þ T Definition 2.4. (Aubin, 1993) We shall say that a function f from X to R fþ1g is lower-continuous at x0 if for all k < f ðx0 Þ, there exists g > 0 such that 8x 2 Bðx0 ; Þ;
k 6 f ðxÞ
ð2:4Þ
We shall say that f is lower semi-continuous if it is lower semi-continuous at every point of X. If f is continuous at x0 , then f is lower-continuous at x0 . T Definition 2.5. (Aubin, 1993) Let f be a function from X to R fþ1g. The sets sðf ; kÞ :¼ fx 2 X jf ðxÞ 6 kg are called lower sections of f.
ð2:5Þ
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T Definition 2.6. (Aubin, 1993) We shall say that a function f from X to R fþ1g is lower semi-compact if all its lower sections are relatively compact. This leads to the following theorem. S Theorem 2.1. (Aubin, 1993) Suppose that a strict function f from X to R fþ1g is both lower semi-continuous and lower semi-compact. Then the set M of elements at which f attains its minimum is non-empty and compact. Proposition 2.1. Assume K 1 ¼ fxjhðxÞ ¼ ag; K 2 ¼ fxjhðxÞ P ag; K 3 ¼ fxjhðxÞ 6 ag where hðxÞ is a linear continuous function of x. Then (a) If K 1 –;, then set K 1 is closed. (b) If K 2 –;, then set K 2 is closed. (c) If K 3 –;, then set K 3 is closed. Proof. The proofs of the second and third assertions are similar to the first one. We shall see how to generalize the first assertion. We now consider sequence of elements xn 2 K 1 converging to x. If set K 1 is open, then x R K 1 , that is, hðxi Þ–hðxÞ. Thus, khðxi Þ hðxÞk ¼ qi > 0. let q0 ¼ min qi and for and e < q0 , e > 0, SðxÞ is a neighborhood of x satisfying that khðxi Þ hðxÞk 6 e < q0 , 8x 2 SðxÞ. Obviously, any element xi of the sequence does not belong to SðxÞ and this contradict that the sequence converges to x. Therefore, K 1 is closed. h S Proposition 2.2. (Aubin, 1993) The functions f ; g from X to R fþ1g are assumed to be lower semi-continuous. Then (a) f þ g is lower semi-continuous. (b) af is lower semi-continuous for any a 2 Rþ . The traditional solution methods require conversion of the chance constraints to their respective equivalents. As we know, this process is usually hard to perform and only successful for some special cases. Let us consider the following form of chance constraint Pfgðx; nÞ 6 0g P b
ð2:6Þ
where n is a stochastic vector. Theorem 2.2. (Liu, 2002) Assume that the stochastic vector n degenerates to a random variable n with distribution function U, then (a) if gðx; nÞ has the form gðx; nÞ ¼ f ðxÞ n, then Pfgðx; nÞ 6 0g P b if and only if hðxÞ 6 K b where K b is the maximal number such that PfK b 6 ng P b. (b) if gðx; nÞ has the form gðx; nÞ ¼ n f ðxÞ, then Pfgðx; nÞ 6 0g P b if and only if hðxÞ P K b where K b is the minimum number such that PfK b P ng P b. According to Proposition 2.1, if f ðxÞ is linear and continuous function of x, sets fxjP fgðx; nÞ ¼ f ðxÞ n 6 0g P bg and fxjP fgðx; nÞ ¼ n f ðxÞ 6 0g P bg are closed if they are non-empty, respectively. 3. Problem description Consider a deterministic supply chain network S ¼ ðN; AÞ, where N is the set of nodes and A is the set of arcs. The set S N consists of the set of suppliers S, the set of processing facilities P and the set of customers C, S i.e., N ¼ S F C. The facilities include manufactories M, finishing facilities (F) and wareS processing S houses W, i.e., P ¼ M F W. Let K be the set of products flowing through the supply chain. For simplicity, we assume that there is only single material and single component needed for production or processing. The following notation is used in the formulation of the model.
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ci :the fixed cost for opening and operating facility i 2 P tij : the per-unit cost of transporting product on arc ðijÞ 2 A qj : per-unit of material or component required to produce one unit of product rj : the per-unit processing requirement for node j mi : the capacity of facility i 2 P si : the capacity of supplier i 2 S for product d j : product demand of customer j Decision variables are: yi ¼
1 0
if facility i 2 P is opened otherwise
xij : the row material of product or product from node i to node j where ðijÞ 2 A. In terms of the above notation, we are ready to formulate a programming model for our supply chain design problem. X X min ci y i þ tij xij i2P
s:t:
ðijÞ2A
8P P xij qj xjl ¼ 0; 8j 2 P > > > i2N l2N > > P > > xij P d j ; 8j 2 C > > > > < i2N P xij 6 si ; 8i 2 S > j2N > > P > > rj xij 6 mj y j ; 8j 2 P > > > > i2N > > : jPj x 2 RjAj þ y 2 Y # f0; 1g
ð3:1Þ
In the above model the objective function minimizes total costs made of: the costs associated with opening and operating the facilities and the operational or transporting costs. The first constraint set enforces the flow conservation of product across any node j 2 P. The second constraint set ensures that the total flow of product to customer j satisfies the demand d j at least. The third constraint set guarantees that the total flow of product from supplier j is less that the supply d j . Finally, the last constraint set enforces the non-negativity restrictions on the corresponding decision variables and the integrality restrictions on the binary variables. It also includes the set Y and X of logical dependencies and restrictions. For simplicity, we have the following compact notation for model (3.1): min
s:t:
cT y þ t T x 8 Nx ¼ 0 > > > > > > < Dx 6 d Sx P s > > > Rx 6 my > > > : jPj x 2 RjAj þ ; y 2 Y # f0; 1g
ð3:2Þ
where vectors c, t, d, and s are corresponding to opening and operational costs, transporting costs, demands, supplies, respectively. The matrices N ; D, and S are appropriate matrices corresponding to the summations on the left-hand-side of the expressions (3.1), respectively. R corresponds to a matrix of rj , and m corresponds to a matrix of mj . To extend the above model to a random fuzzy setting, we assume that transportation costs, demands, and ^ ^sÞ represents the random fuzzy vector. The resultsupplies are random fuzzy variables. In particular, n ¼ ð^t; d; ing formulation is as followed:
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min s:t:
cT y þ ^tT x 8 ^ > < Dx 6 d Sx P ^s > : jPj ðx; yÞ 2 G0 ¼ fx 2 RjAj þ ; y 2 Y # f0; 1g jNx ¼ 0; Rx 6 myg
ð3:3Þ
which is a random fuzzy programming. 4. Solution method Considering problem (3.3). In practice, it would certainly be more appropriate to consider that the possible values of the fuzzy parameters in the probability distributions usually involve the ambiguity of the experts’s understanding of the real system. For this reason, in this paper, we consider the following auxiliary programming problem involving random fuzzy variables: min cT y þ E gTt x 8 PfDj x P gd j g P bd j ; 8j 2 C > > > < PfS x 6 g g P b ; 8i 2 S ð4:1Þ i si sj s:t: > ðx; yÞ 2 G0 > > : gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ with 0 6 bd j 6 1, 0 6 bsi 6 1, and Dj and S i indicating the j-th and i-th of D and S, respectively. Note that the first and second constraint sets are disjoint chance constraints. The above model is an (a; b) chance constrained programming. It should be emphasized here that the parameters gn ¼ ðgt ; gd ; gs Þ are treated as decision variables rather than constants. On the basis of the a-cut sets of the random fuzzy variables, we introduce the concept of a-optimal solutions to it: ¼ fxjPfDj x P gd j g P bd j ; 8j 2 C; g, G2 ðgs Þ ¼ Definition 4.1. (a-optimal solution) Set T G1 ðgd Þ T fxjPfS i x 6 gsi g P bsi ; 8i 2 S; g and G1 ðgd gs Þ ¼ G0 G1 ðgd Þ G2 ðgs Þ. A feasibility solution ðx ; y Þ of (4.1) is T T said to be an a-optimal solution, if and only if there exists cT y þ gT t x 6 c y þ gt x for any ðx; yÞ 2 Gðgd ; gs Þ; gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ where the corresponding values of parameters g ¼ ðgt ; gd ; gs Þ are called a-level optimal parameters. 4.1. The existence of solutions Let we consider the following problem: min cT y þ E gTt x 8 > < PfDj x P gd j g P bd j ; 8j 2 C PfS i x 6 gsi g P bsj ; 8i 2 S s:t: > : ðx; yÞ 2 G0 where gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ. According to Theorem 2.2, problem (4.2) is equivalent to min cT y þ E gTt x 8 Dj x P minfK b jPfK b P gd j g P bd j g; 8j 2 C > Kb > < S i x 6 maxfK b jPfK b P gsi g P bsi g; 8i 2 S s:t: Kb > > : ðx; yÞ 2 G0
ð4:2Þ
ð4:3Þ
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We now show that the feasibility set Gðgd ; gs Þ is bounded and closed. Assume that G0 , G1 ðgd Þ, G2 ðgs Þ–0. jPj According to x 2 RjAj þ ; y 2 Y # f0; 1g , Gðgd ; gs Þ is bounded. According to Proposition 2.1, Gðgd ; gs Þ is closed. That is the feasibility set Gðgd ; gs Þ is compact. Proposition 4.1. Let f be a function from Gðgd ; gs Þ to R and f ¼ cT y þ E gTt x x. We set f; if ðx; yÞ 2 Gðgd ; gs Þ fGðgd ;gs Þ ðx; yÞ :¼ þ1; if ðx; yÞ 2 ðX ; Y Þ n Gðgd ; gs Þ jAj jPj where S ðX ; Y Þ ¼ Rþ Rþ and fGðgd ;gs Þ ðx; yÞ is no longer a real-valued function but a function from ðX ; Y Þ to R fþ1g. Then fGðgd ;gs Þ ðx; yÞ is lower semi-continuous. jPj Proof. Firstly, we consider f1 ¼ cT y. The domain of function f1 is Dom f 1 :¼ y 2 RjPj þ jf1 < þ1 ¼ f0; 1g . jPj We take k 2 R, then Sðf1 ; kÞ :¼ y 2 Rþ jfS1 6 k # Dom f 1 . Obviously, anarbitrary section Sðf1 ; kÞ is closed. fþ1g, is lower semi-continuous. Next, we consider f2 ¼ E gTt x Thus, f1 ¼ cT y, which is from RjPj þ Sto R 2.1, an arbitrary section which is from RþjAj to R fþ1g. According to Proposition S jAj T jf < þk is closed. It follows that f ¼ E g to R fþ1g, is lower x which is from R Sðf2 ; kÞ :¼ x 2 RjAj 2 2 t þ þ semi-continuous. Resulted from Proposition 2.2, fGðgd ;gs Þ ðx; yÞ is lower semi-continuous. h
apply Theorem 2.1 to the function fGðgd ;gs Þ ðx; yÞ defined by fGðgd ;gs Þ ðx; yÞ ¼ cT yþ We jPj E gTt x if ðx; yÞ 2 Gðgd ; gs Þ and fGðgd ;gs Þ ðx; yÞ ¼ þ1; if x 2 ðX ; Y Þ n Gðgd ; gs Þ where ðX ; Y Þ ¼ RjAj þ Rþ , noting that fGðgd ;gs Þ ðx; yÞ is lower semi-continuous (see Proposition 4.1), that fGðgd ;gs Þ ðx; yÞ is lower semi-compact, and Gðgd ; gs Þ is relative compact. Solutions of problem (4.3) and so (4.2) exist. As is well known, a-cuts of each fuzzy number are closed intervals of real numbers for all a 2 ð0; 1, which is an important property of the fuzzy number. Obviously, the optimal solutions of problem (4.1) exist. 4.2. Deterministic equivalent In the paper, the demands and capacities of suppliers, have lognormal distributions with known scale parameters and unknown location parameters. Consider the case of the demand d j , having a lognormal distribution of which the mean value M d j is a triangular fuzzy number and the standard deviation were chosen as certain fractions of the mean value. Similarly, the capacity of supplier si , has a lognormal distribution of which the mean value M si is a triangular fuzzy number and the standard deviation were chosen as certain fractions of the mean value. Now from the properties of the a-level sets for the vectors of the fuzzy numbers M d j and hM si , it should i be noted that the a-level sets for M d j and M si can be denoted, respectively by the intervals M Ldj a ; M Rdj a and h i L M si a ; M Rsi a . The corresponding random variable intervals are gLdj a ; gRdj a and gLsi a ; gRsi a . For a given bd j , if M 1d j a 6 M 2d j a , k 1bd 6 k 2bd ; For a given bsi , if M 1si a 6 M 2si a , k 1bs 6 k 2bs . According to Theorem 2.1 the first and secj
j
i
i
ond constraint set of (4.3) are converted to Dj x P k bd j , S i x 6 k bsi , respectively, where Pfk bd j P gd j g ¼ bd j and Pfk bsi P gsi g ¼ bsi . Now we introduce the following set-valued functions. T dj ðk bd j Þ ¼ fxjd j x P k bd j g; T si ðk bsi Þ ¼ fxjS i x 6 k bsi g;
8j 2 C 8i 2 S
ð4:4Þ ð4:5Þ
Then, it can be verified that the following relations hold for T dj ðÞ and T si ðÞ, when x P 0. If k 1bd 6 k 2bd , then j j T dj ðk 1bd Þ T dj ðk 2bd Þ; If k 1bs 6 k 2bs , then T si ðk 1bs Þ T dj ðk 2bs Þ. Consequently, we can obtain an optimal solution to j j i i i i (4.1) by solving the following problem.
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min s:t:
T cT y þ M Tt 8 D x P k Lbd > > j < j R S x P k i b si > > : ðx; yÞ 2 G0
ð4:6Þ
where Pfk Lbd P gLdj g ¼ bd j and Pfk Rbs P gRsi g ¼ bsi . j i Then it is interesting to note that M Lt , k Lbd and k Rbs are a-level optimal parameters for any a-optimal solution, j i The relations between the optimal solutions to problem (4.6) and the a-optimal solution to (4.1) can be characterized by the following theorems. Theorem 4.1. If ðx ; y Þ is a optimal solutions to (4.6), then ðx ; y Þ is an a-optimal solution to the (4.1). Proof. Assume that ðx ; y Þ is not an a-optimal solution to the (4.1). Then, since M Lt , k Lbd and k Rbs are a-level j i optimal parameters to the (4.1), there exist ðx; yÞ 2 Gðgd ; gs Þ and gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ such that x . Then it holds that cT y þ E gTt x < cT y þ E gT t T T T LT cT y þ M LT t x 6 c y þ Mt x < c y Mt x
which contradicts the fact that ðx ; y Þ is a optimal solution to (4.6).
h
Theorem 4.2. If ðx ; y Þ is a optimal solution to (4.1), then ðx ; y Þ is an a-optimal solution to the (4.6). Proof. Assume that ðx ; y Þ is not an a-optimal solution to the (4.1). Since M Lt , k Lbd and k Rbs are a-level optimal j i parameters to the (4.1), according to Definition 4.1, ðx ; y Þ is a optimal solution to (4.6). h Obviously, problem (4.6) is the deterministic equivalent to the problem (4.1), when gd j ; 8j 2 C and gsj ; 8i 2 S follow lognormal distributions. At a given possibility level a, we can convert problem (4.1) to the formulation (4.6) which is a deterministic mixed-integer 0–1 programming. 5. st-GA Recently, there has been an increasing interest in using various evolutionary computation methods to solve hard optimization problems (Gen & Cheng, 1997). Among them, genetic algorithm (GA) is the one of the most well known class of evolutionary algorithms. The underlying principles of GAs were first exposed by Holland (1975). GAs deal with a coding of the problem instead of decision variables. They require no domain knowledge—only the payoff or objectives for evaluating fitness after operating genetic operations. In addition, traditional methods use deterministic transition rules to guide the search, such as hill-climbing, neighborhood search. Another difference between traditional methods and GAs is the latter searches from a set of points, while the former from a single point. These makes GAs more robust than traditional methods regarding their potential as optimization techniques to solve many real world problems (Gen & Cheng, 1997). Michalewicz (1994) was the first researcher, who used GA for solving linear and non-linear transportation/ distribution problems. In his method, he represented each chromosome of the problem by using m n matrix. The use of st-GA for solving some network problem was introduced and applied by Gen and Cheng (1997), and Syarif, Yun and Gen (2002). They employed Pru¨fer number in order to represent a candidate solution to the problems and developed feasibility criteria of Pru¨fer number to be decoded into a spanning-tree. The verification for the excellence of the Pru¨fer number was addressed by Zhou and Gen (1996). They noted that the use of Pru¨fer number is more suitable for encoding a spanning-tree, especially in some research fields, such as: transportation problems, minimum spanning-tree problems, and so on. Also, it is shown that we can use only m þ n 2 digit number to uniquely represent a distribution network with m suppliers and n destinations, where each digit is an integer between 1 and m þ n. This means that Pru¨fer number representation is more efficient in the sense of required memory for computation than that of matrix-based representation. In this paper, we consider the network problem (4.6). As the solution method, we use st-GA and design feasibility check and develop repairing procedure, so that it can be applied for relatively large size problems. The steps of our algorithm for solving the problem are listed as follows:
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Step 1. Set the initial values and the parameters of genetic algorithms: population size pop_size, crossover rate pc , and maximum generation maxgen. Step 2. Generate the initial population. Step 3. Generic operate: crossover and mutation. Step 4. Check the feasibility of the offspring and repair the infeasible offspring. Step 5. Evaluate and select the chromosomes. Step 6. Repeat the second to fifth steps for the given number of maxgen. Step 7. Report the best chromosome as the optimal solution.
5.1. Representation and feasibility of the chromosome For this problem, we use sub-tree S P , sub-tree P W , sub-tree W C to represent the distribution pattern for suppliers to plant, plant to warehouses, and warehouses to customers, respectively. Each chromosome consists of five parts. The first part is Pl binary digits to represent the opened/closed plants. The second part is Wh binary digits to represent opened/closed warehouses. The last three parts are three Pru¨fer numbers representing the distribution pattern of each stage, respectively. In the first step which generates the chromosome, we generate the two 0–1 variables and check the total capacities of open plants and open warehouses to satisfy all the customer demands. If the total capacity is less than the total demand, then open the closed facility with maximum capacity until the total capacity of facilities is enough to satisfy the customer demands. Generating the three Pru¨fer numbers is followed. For the sub-tree W C, denote the suppliers set O ¼ f1; 2; . . . ; W g, and plants set D ¼ fW þ 1; W þ 2; . . . ; W þ Cg. Obviously, this distribution has W þ C nodes, which means that we need W þ C 2 digits Pru¨fer number in the range ½1; W þ C to uniquely represent the sub-tree W þ C. 5.1.1. Feasibility check for Pru¨fer number In the Pru¨fer number P ðT Þ, let P ðT Þ be the set of all nodes not included in P ðT Þ. Ri denotes the number of appearances of node i 2 P ðT Þ and Li denotes the number of connections of node i 2 O [ D. The Pru¨fer number P ðT Þ is said to be feasible if X X Li ¼ Li ð5:1Þ i2O
i2D
The feasibility check and repairing procedure for the Pru¨fer number is as follows: Step 1. Compute Ri for i 2 O [ D in P ðT Þ Step 2. Li P ¼ Ri þ 1 P Step 3. If i2O Li > i2D Li , then select one digit in P ðT Þ which contains node i 2 O and replace it with the number j 2 D generated randomly. Otherwise, select one digit in P ðT Þ which contains node i 2 D and replace it with the number j 2 O generated randomly. After checking for the feasibility, the Pru¨fer number can be decoded into spanning-trees in order to determine the distributaries pattern for the network. The procedures of decoding Pru¨fer number is given as follows: Step 1. Let i be the smallest label node of P ðT Þ and j be the leftmost digit of P ðT Þ. Step 2. If i and j are not in the same set O or D, add the edge from i to j into the tree. Otherwise, select the next digit k from P ðT Þ that not included in the same set with i. Exchange j with k, add the edge ði; kÞ to the tree. Step 3. Remove j(or k) form P ðT Þ and i from P ðT Þ. If j(or k) does not appear anywhere in the remaining part of P ðT Þ, put it into P ðT Þ. Designate i as no longer eligible. Step 4. Assign the available amount of units to xxj ¼ minfai ; bj g or xij ¼ minfai ; bk g. Step 5. Update the availability ai ¼ ai xij and bj ¼ bj xij Step 6. Repeat the process Step 1–Step 5, until no digits left in P ðT Þ.
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Step 7. If no digits remain in P ðT Þ, there are exactly two nodes, r and s, still negligible for consideration. Add link from r to s into the tree and form a tree with W þ C 1 links. Step 8. If no available amount of units to assign, then stop. Otherwise, there are remaining supply of node r and demand of node s, add edge (r; s) to tree and assign the available amount of units xrs ¼ minfar ; bs g to edge. If there exists a cycle, then remove the edge that assigned zero flow. By using similar ways, we can generate two other Pru¨fer numbers representing P W and S P sub-trees. These two Pru¨fer numbers have P þ W 2 and S þ P 2 digits, respectively. Note that in order to ensure the flow conservation of product across any node, we first decode the Pru¨fer number W C, then P W and finally S P . When decoding the chromosome, firstly update the capacity of open facilities by their total exporting amount to the next stage and change the capacity of the closed facilities to be zero. 5.2. Genetic operations The genetic algorithm uses the individuals in the current generation to create the children that make up the next generation. Besides elite children, which correspond to the individuals in the current generation with the best fitness values, the algorithm creates crossover children by selecting vector entries, or genes, from a pair of individuals in the current generation and combines them to form a child and mutation children by applying random changes to a single individual in the current generation to create a child. Both processes are essential to the genetic algorithm. Crossover enables the algorithm to extract the best genes from different individuals and recombine them into potentially superior children. Mutation adds to the diversity of a population and thereby increases the likelihood that the algorithm will generate individuals with better fitness values. Without mutation, the algorithm could only produce individuals whose genes were a subset of the combined genes in the initial population. 5.2.1. Crossover The crossover is done by exchanging the information of two parents to provide a powerful exploration capability. In this paper, we employ a one-cut-point and exchanges the right parts of two parents to generate offspring. However, before decoding all Pru¨fer numbers, we still need to check whether the capacity of the opened facilities is enough to satisfy the demand or not, and check the feasibility of the Pru¨fer numbers. Fig. 2 shows the one-point crossover process. 5.2.2. Mutation Mutation adds to the diversity of a population and thereby increases the likelihood that the algorithm will generate individual with better fitness values. Without mutation, the algorithm could only produce individuals whose genes were a subset of the combined genes in the initial population. We use inversion mutation operations. The inversion mutation selects two positions within a chromosome at random and then inverts the sub-string between these two positions. This mutation operators will generate
Fig. 2. The one-point crossover process.
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Fig. 3. The inversion mutation process.
feasible Pru¨fer numbers if the parents are feasible. However, we also need to check the feasibility of the opened facilities to satisfy the demand. Fig. 3 shows the illustration of those mutation processes. We can specify how many of each type of children the algorithm creates as follows: (a) Elite count, in reproduction options, specifies the number of elite children. (b) Crossover fraction, in reproduction options, specifies the fraction of the population, other than elite children, that are crossover children. For example, if the population size is 20, the elite count is 2, and the crossover fraction is 0.8, the numbers of each type of children in the next generation are as follows: There are 2 elite children and 18 individuals, so the algorithm rounds 0.8 18 = 14.4 to 14 to get the number of crossover children. The remaining 4 individuals, other than elite children, are mutation children. 6. Computational results In this section we describe numerical example using the proposed methodology and st-GA for solving a realistic multi-stage SC design problems, then demonstrate the computational effectiveness afforded by our method. Our test problem is that of a domestic supply chain network for a company. Henceforth this problem is referred to as the ‘‘domestic” problem. The supply chain topology with all possible center locations and transportation channels for the domestic problem is presented in Fig. 4. The main characteristics of the networks are presented in Table 1. We develop fuzzy mean values for the random fuzzy parameters from the values for the operational or transportational cost, demand and supply capacity parameters from Shapiro (2001). The standard deviations 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 supplier plant warehouse customer
0.2 0.1
0
0.2
0.4
0.6
0.8
1
1.2
Fig. 4. The distribution graph of hospitals and clinics.
1.4
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Table 1 Supply chain network characteristics.
Total facilities Supplies Plants Warehouses Customers Transportation channels
Domestic
Size of the problem
27 3 10 17 230 4110
Constraints Binary variables Total variables Coefficients
288 27 4137 12584
Table 2 Distribution pattern of the first and second stage. Suppliers (capacity)
Plants 1
2
3
4
5
6
7
8
9
10
1(14735) 2(13237) 3(14543)
4579 – –
– 2893 –
– 3707 –
– 3515 –
– – 3775.8
– – –
– – 3387
– 2451 –
– – –
– – 2444.3
Plant capacity Plants
4579 2893 Warehouses
3707
3515
4563
4172
3387
2451
4245
3192
1 2 3 4 5 6 7 8 9 10 Warehouse Capacity
1
2
3
4
5
6
7
8
9
10
11
– – 2426.1 – – – – – – 2144.5 – – – – – – – – – –
– – – – – – – – – – – – – – – – – – – –
– – – – – 1466.3 – – – – – – 2730 – – – – – – –
261.7 2255.9 – – – – – – – – – – – – 2451 – – – – –
– 891.3 – – – – – – – – – – – – – – – – – –
– – – – – 864.6 – 998.3 – – – – – – – – – – 2444.3 –
–
1170
–
–
–
–
466.9
–
–
–
–
–
1376.1
–
–
–
2516.7
–
–
–
– 29.2 –
750.2
–
851.9
–
–
–
–
–
–
–
–
657
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
2426.1 2519.7
2351.1 2586.3
2730 2566.6
2712.7 2255.9
2708.6 2525.4
2444.3 2614.9
2516.7
2387.1
2725.6
2350.3
2390.6
for the distributions were chosen as certain fractions of the mean values. The linear and mixed-integer programming problem (4.6) was solved by st-GA, which is implemented using Matlab7.0 program language. All computations were carried out on a Pentium IV 2.4GHZ PC running Windows XP with GA parameters pop-size = 100, P c ¼ 0:8 and elite count = 2. One heuristic distribution pattern solution of the first and second stage of this case study is shown in Table 2. It is shown that our methodology and st-GA is effective. However, since the search apace of this kind of problem is so large, it is very important to set the experiment with reasonable population size and maximum generation in order to ensure that we have a good results. More alternatives can be generated by varying the values of a and b.
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7. Conclusions Multifold uncertain programming is a growing subject. Here we provided some further researches in the random fuzzy programming area, which is an important branch of multifold uncertain programming. For the mathematical properties, we considered the methodology of a class of random fuzzy programming model, deterministic equivalent, and solution existence. Other aspects, such as sensitivity analysis, dual theorems are waited to investigated. From the computational viewpoint, we developed a st-GA for the deterministic equivalent model which is also used for solving other mixed 0–1 integer mathematic model of this class. From the applied viewpoint, we considered the random fuzzy environment in multi-stage SC design and applied our methodology to the random fuzzy mathematic model presented. Random fuzzy programming may be applied to any optimization problems with random fuzzy factors, for example, queuing system, finance, energy system. Acknowledgements This research was supported by the National Science Fund for Distinguished Young Scholars, PR China (Grant No. 70425005), and the Teaching and Research Award Program for Outstanding Young Teacher in Higher Eduction Institutions of MOE of PR China (Grant No. 20023834-3). References Aikens, C. H. (1985). Facility location models for distributing planning. European Journal of Operational Research, 22(3), 263–279. Amiri, A. (2004). Designing a distribution network in a supply chain system: Formulation and efficient solution procedure. European Journal of Operational Research. Aubin, J. P. (1993). Optima and equilibria. Berlein Heidelberg: Springer-Verlag. Charnes, A., & Cooper, W. W. (1959). Chance-constrained programming. Management Science, 6(1), 73–79. Cheung, R. K.-M., & Powell, B. W. (1996). Models and algorithms for distribution problems with uncertain demands. Transportation Science, 30, 822–844. Gen, M., & Cheng, R. (1997). Genetic algorithms and engineering design. New York: Wiley. Geoffrion, A. M., & Graves, G. W. (1974). Multi-commodity distribution system design by benders decomposition. Management Science, 20, 822–844. Geoffrion, A. M., & Powers, R. F. (1995). Twenty years of strategic distribution system design: An evolution perspective. Interfaces, 25, 105–128. Goetschalckx, M., Vidal, C. J., & Dogan, K. (2002). Modeling and design of global logistics systems: A review of integrated strategic and tactical models and design algorithms. European Journal of Operational Research, 143, 1–18. Guille´n, G., Mele, F. D., Bagajewicz, M. J., Espun˜a, A., & Puigjaner, L. (2005). Multiobjective supply chain design under uncertainty. Chemical Engineering Science, 60, 1535–1553. Holland, J. (1975). Adaptation in natural and artificial systems. Ann Arbor: University of Michigan Press. Landeghen, H. V., & Vanmaele, H. (2002). Robust planning: A new paradigm for demand chain planning. Journal of Operations Management, 20, 769–-783. Liu, B. (2002). Theory and practice of uncertain programming. Physica-berlag. Michalewicz, Z. (1994). Genetic algorithms + data structures = evolution programs (2nd ed.). New York: Springer. Min, H., & Zhou, G. (2002). Supply chain modeling: Past, present and future. Computers and Industrial Engineering, 43, 231–249. Nattier, W. (2001). Random fuzzy variables of second order and applications to statistical inference. Information Sciences, 133, 69–88. Owen, S. H., & Daskin, M. S. (1998). Strategic facility location: A review. European Journal of Operational Research, 111, 423–447. Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. European Journal of Operational Research, 167, 96–115. Shapiro, J. F. (2001). Modeling the supply chain. Thomson Learning. Syarif, A., Yun, Y. S., & Gen, M. (2002). Study on multi-stage logistic chain network: A spanning tree-based genetic algorithm approach. Computers & Industrial Engineering, 43, 299–314. Yan, H., Yu, Z., & Cheng, T. C. E. (2003). A strategic model for supply chain design with logical constraints: Formulation and solution. Computers & Operations Research, 30, 2135–2155. Zhou, G., & Gen, M. (1996). A note on genetic algorithms approach to the degree-constrained spanning tree problem. Networks, 30, 105–109. Jiuping Xu was born in Chongqing, China, in 1962. He received the Ph.D. of applied mathematics from Tsinghua University, Beijing, China; and the Ph.D. of physical chemistry from Sichuan University, Chengdu, China, in 1995 and 1999, respectively. He is currently Professor at Sichuan University and the vice-president of The Systems Engineering Society of China. His current research interests are in the areas of systems science and engineering, applied mathematics, and physical chemistry, management science and engineering man-
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agement. He has published 31 books and over 300 journal papers in IEEE Transaction on Fuzzy Systems, Fuzzy Sets and Systems, The International Journal of Management Sciences, Information Sciences, Journal of Computational and Applied Mathematics, Mathematics and Computers in Simulation, and Mathematical and Computer Modelling. Dr. Xu was received the IFORS Prize for Operational Research in Development of International Federation of Operations Research Societies at Vancouver, in 1996. He has received numerous prizes in China, including the China Youth Prize of Science and Technology, in 2004. Yanan He received M.S. degree from Sichuan University, Chengdu, China, in 2006. She is currently working toward the Ph.D. degree at Chinese Academy of Sciences, China. Her main research interests are in the fields of decision making, uncertain programming, and energy economics. Mitsuo Gen received the Ph.D. degree from the Kogakuin University, Japan in 1974. He was Lecturer at Ashikaga Institute of Technology, Japan in 1974–1980; Assoc. Professor at Ashikaga Institute of Technology, Japan in 1987.4–2003.3; He is currently a Professor of Graduate School of Information, Production & Systems, Waseda University. He was Visiting Assoc. Prof. at University of California at Berkeley, USA in 1999–2000. His research interests include Genetic Algorithms, Neural Network, Fuzzy Logic, and the applications of evolutionary techniques to Network Design, Schedule, System Reliability Design,etc. He has authored Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York (2000) with Dr. Runwei Cheng. Dr.Gen is a member of the editorial board of several international journals, such as, Computational Intelligence of Computers and Industrial Engineering, OR Spectrum, Fuzzy Optimization and Decision Making, International Journal of Manufacturing Technology & Management and International Journal of Smart Engineering & Systems Design.
Computers & Industrial Engineering 56 (2009) 937–950 www.elsevier.com/locate/caie
A class of random fuzzy programming and its application to supply chain design Jiuping Xu a,*, Yanan He b, Mistsuo Gen c a
Uncertainty Decision-Making Laboratory, School of Business and Administration, Sichuan University, Chengdu 610064, China b Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China c Graduate School of Information, Production and Systems, Waseda University, Kitakyushu 808-0315, Japan Available online 14 October 2008
Abstract Supply chain design problems have recently raised a lot of interest since the opportunity of an integrated management of the supply chain can reduce the propagation of undesirable events through the network and can affect decisively the profitability of the members. Often uncertainties may be associated with demand, supply or various relevant costs. In most of the existing models uncertainties are treated as randomness and are handled by appealing to probability theory. However, in certain situations uncertainties are due to random fuzziness and in these case the random fuzzy programming is applicable, while no attempt has been made to define those uncertain parameters as random fuzzy variables. We propose a random fuzzy programming model and a methodology for solving a multi-stage supply chain design problem of a realistic scale in the random fuzzy environment. Based on the concept of random fuzzy variable, we use probability theory, fuzzy set theory and optimizing theory as our research tool. At a given possibility a, we provide an auxiliary programming model of the random fuzzy programming and convert it to a deterministic mixed 0–1 integer programming model, of which the optimal solutions are proved to exist. Then we present a novel technique called spanning-tree based on genetic algorithm (st-GA) to get the heuristic solutions. A computational example involving a relatively large supply chain is presented to highlight the significance of the random fuzzy model as well as the effectiveness of the proposed solution method. Random fuzzy variable is a very useful tool to describe the uncertain parameters and our methodology may be applied to any optimization problems with random fuzzy factors. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Random fuzzy variable; Random fuzzy programming; Supply chain design; st-GA
1. Introduction A supply chain is a network comprised of a set of geographically dispersed facilities (suppliers, plants, and warehouses or distribution centers) where raw materials, intermediate products, or finished products are acquired, transformed, stored, or sold. It is often regarded as the art of bringing the right amount of the right
*
Corresponding author. Tel.: +86 28 85418522; fax: +86 28 85400222. E-mail addresses: [email protected], [email protected] (J. Xu), [email protected] (Y. He).
0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.09.045
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product to the right place at the right time. If the facilities are to distribute product directly to customers, then single-stage model is appropriate. On the other hand, if several facilities are to be sited between the suppliers to the customers in order to produce product or act as regional warehouses or distribution centers, then multistage model is the appropriate model. The structure of the aforementioned multi-stage supply chain is depicted in Fig. 1. Mathematical programming models have proven their usefulness as analytical tools to optimize complex decision-making problems such as those encountered in supply chain planning. Geoffrion and Graves (1974) described a multi-commodity distribution system design problem and solved it by Benders Decomposition. This is probably the first paper that presents a comprehensive MIP model for the strategic design of supply chain networks. After that, a diversity of deterministic mathematical programming models dealing with the design of supply chain networks can be found in the literature. See for example Aikens (1985), Goetschalckx, Vidal, and Dogan (2002), Geoffrion and Powers (1995), Yan, Yu, and Cheng (2003), and Amiri (2004). Under the most circumstances, the critical design parameters for the supply chain, such as customer demands, prices and resource capacities are generally uncertain. Moreover, the arrival of regional economic alliances, for instance the Asian Pacific Economic Alliance and the European Union, have prompted many corporations to move more and more towards global supply chains, and therefore to become exposed to risky factors such as exchange rates, reliability of transportation channels, and transfer prices (Santoso, Ahmed, Goetschalckx, & Shapiro, 2005). Unless the supply chain is designed to be robust with respect to the uncertain operating conditions, the impact of operational inefficiencies such as delays and disruptions will be larger than necessary. Uncertain supply chain design has been one of the promising subjects. Beginning with the seminal works of Charnes and Cooper (1959), optimization under uncertainty has experienced rapid development in both theory and algorithms. A big amount of stochastic programming models have been proposed for strategic and tactical planning. See for example Owen and Daskin (1998), Cheung and Powell (1996), Guille´n, Mele, Bagajewicz, Espun˜a, and Puigjaner (2005), Min and Zhou (2002), and Landeghen and Vanmaele (2002). However, in certain situations, the assumption of precise parameters of probability distributions is seriously questioned. The parameters are fixed statistically estimated using past demand information, while demand does not stay ‘static’ in fact. When the conditioning variables, such as the technologic innovations and preferences of consumers, considerably change, the mean and variance of the demand distribution are possible to change. Besides, it is almost impossible to specify exactly the true values to the parameters, especially in the absence of abundant information as in the case of demand of new products. It is more appropriate to estimate the parameters ‘‘approximately a”. Thus, based on expert experience fuzzy variables are considered to describe them. In this case, random variables with imprecise parameters are random fuzzy variables. By random fuzzy programming we mean the optimization theory in random fuzzy environment. Random fuzzy programming makes the supply chain design plan more flexible when the parameters of the coefficients’
Fig. 1. Supply chain structure.
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distribution are uncertain. In practice, the values of the fuzzy parameters can be obtained according to the expert experience. Different numbers of the fuzzy variables reflect different conditions which affect the parameters of the probability distribution. The concept of random fuzzy variables was provided by Liu (2002), which is different from the definition used by Nattier (2001). To the best of our knowledge, there is a little research for the programming and solving supply chain design problems in random fuzzy environment. Therefore, there is strong motivation for further research in the area. We wish to highlight the significance of introducing random fuzziness in supply chain design modeling. In this paper, we consider a multi-stage supply chain design problem with a number of potential facilities (plants, warehouses) and the task includes the choices of facilities to be opened and the distribution pattern design to satisfy the demand with minimum cost in random fuzzy environment. A random fuzzy programming model is proposed and a methodology is developed. The remainder of this paper is organized as follows: in the next section, we state some basic concepts and results on random fuzzy variables and optimizing theory. In Section 3, we develop a random fuzzy mathematical programming model for a multi-stage supply chain designing problem in a random fuzzy environment. In Section 4, we present our methodology, and discuss the existence of solution. In Section 5, we introduce the stGA to get the heuristic solutions. Section 6 provide a real-life multi-stage supply chain design test case. Finally we offer some concluding remarks in Section 7. 2. Preliminaries In this section, we will state some basic concepts and theorems on random fuzzy variables and optimizing theory. These results are crucial for the remainder of this paper. Interested readers may consult Liu (2002) where important properties of random fuzzy variables are recorded. Definition 2.1. (Liu, 2002) A random fuzzy variable is a function from the possibility space (H; PðHÞ; Pos) to a collection of random variables R, where H is a universe, PðHÞ is the power of H, and Pos is a possibility measure defined on PðHÞ. Definition 2.2. (Liu, 2002) Let n be a random fuzzy variable on the possibility space (H; PðHÞ; Pos). Then its membership function is derived from the possibility measure Pos by lðgÞ ¼ Posfh 2 HjnðhÞ ¼ gg;
g2R
ð2:1Þ
The a-cut na is defined as na ¼ fg 2 Rjln ðgÞ P ag
ð2:2Þ
Let an n-dimensional random fuzzy vector n ¼ ðn1 ; n2 ; . . . ; nn Þ. Then we can introduce the following a-level set or a-cut of n. Definition 2.3. The a-level (a 2 ½0; 1) set of the random fuzzy vector n ¼ ðn1 ; n2 ; . . . ; nn Þ is defined as the ordinary set La ðnÞ: La ðnÞ ¼ fg ¼ ðg1 ; g2 ; . . . ; gn Þjln1 ðg1 Þ P a; ln2 ðg2 Þ P a; . . . ; lnn ðgn Þ P ag ð2:3Þ T Definition 2.4. (Aubin, 1993) We shall say that a function f from X to R fþ1g is lower-continuous at x0 if for all k < f ðx0 Þ, there exists g > 0 such that 8x 2 Bðx0 ; Þ;
k 6 f ðxÞ
ð2:4Þ
We shall say that f is lower semi-continuous if it is lower semi-continuous at every point of X. If f is continuous at x0 , then f is lower-continuous at x0 . T Definition 2.5. (Aubin, 1993) Let f be a function from X to R fþ1g. The sets sðf ; kÞ :¼ fx 2 X jf ðxÞ 6 kg are called lower sections of f.
ð2:5Þ
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T Definition 2.6. (Aubin, 1993) We shall say that a function f from X to R fþ1g is lower semi-compact if all its lower sections are relatively compact. This leads to the following theorem. S Theorem 2.1. (Aubin, 1993) Suppose that a strict function f from X to R fþ1g is both lower semi-continuous and lower semi-compact. Then the set M of elements at which f attains its minimum is non-empty and compact. Proposition 2.1. Assume K 1 ¼ fxjhðxÞ ¼ ag; K 2 ¼ fxjhðxÞ P ag; K 3 ¼ fxjhðxÞ 6 ag where hðxÞ is a linear continuous function of x. Then (a) If K 1 –;, then set K 1 is closed. (b) If K 2 –;, then set K 2 is closed. (c) If K 3 –;, then set K 3 is closed. Proof. The proofs of the second and third assertions are similar to the first one. We shall see how to generalize the first assertion. We now consider sequence of elements xn 2 K 1 converging to x. If set K 1 is open, then x R K 1 , that is, hðxi Þ–hðxÞ. Thus, khðxi Þ hðxÞk ¼ qi > 0. let q0 ¼ min qi and for and e < q0 , e > 0, SðxÞ is a neighborhood of x satisfying that khðxi Þ hðxÞk 6 e < q0 , 8x 2 SðxÞ. Obviously, any element xi of the sequence does not belong to SðxÞ and this contradict that the sequence converges to x. Therefore, K 1 is closed. h S Proposition 2.2. (Aubin, 1993) The functions f ; g from X to R fþ1g are assumed to be lower semi-continuous. Then (a) f þ g is lower semi-continuous. (b) af is lower semi-continuous for any a 2 Rþ . The traditional solution methods require conversion of the chance constraints to their respective equivalents. As we know, this process is usually hard to perform and only successful for some special cases. Let us consider the following form of chance constraint Pfgðx; nÞ 6 0g P b
ð2:6Þ
where n is a stochastic vector. Theorem 2.2. (Liu, 2002) Assume that the stochastic vector n degenerates to a random variable n with distribution function U, then (a) if gðx; nÞ has the form gðx; nÞ ¼ f ðxÞ n, then Pfgðx; nÞ 6 0g P b if and only if hðxÞ 6 K b where K b is the maximal number such that PfK b 6 ng P b. (b) if gðx; nÞ has the form gðx; nÞ ¼ n f ðxÞ, then Pfgðx; nÞ 6 0g P b if and only if hðxÞ P K b where K b is the minimum number such that PfK b P ng P b. According to Proposition 2.1, if f ðxÞ is linear and continuous function of x, sets fxjP fgðx; nÞ ¼ f ðxÞ n 6 0g P bg and fxjP fgðx; nÞ ¼ n f ðxÞ 6 0g P bg are closed if they are non-empty, respectively. 3. Problem description Consider a deterministic supply chain network S ¼ ðN; AÞ, where N is the set of nodes and A is the set of arcs. The set S N consists of the set of suppliers S, the set of processing facilities P and the set of customers C, S i.e., N ¼ S F C. The facilities include manufactories M, finishing facilities (F) and wareS processing S houses W, i.e., P ¼ M F W. Let K be the set of products flowing through the supply chain. For simplicity, we assume that there is only single material and single component needed for production or processing. The following notation is used in the formulation of the model.
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ci :the fixed cost for opening and operating facility i 2 P tij : the per-unit cost of transporting product on arc ðijÞ 2 A qj : per-unit of material or component required to produce one unit of product rj : the per-unit processing requirement for node j mi : the capacity of facility i 2 P si : the capacity of supplier i 2 S for product d j : product demand of customer j Decision variables are: yi ¼
1 0
if facility i 2 P is opened otherwise
xij : the row material of product or product from node i to node j where ðijÞ 2 A. In terms of the above notation, we are ready to formulate a programming model for our supply chain design problem. X X min ci y i þ tij xij i2P
s:t:
ðijÞ2A
8P P xij qj xjl ¼ 0; 8j 2 P > > > i2N l2N > > P > > xij P d j ; 8j 2 C > > > > < i2N P xij 6 si ; 8i 2 S > j2N > > P > > rj xij 6 mj y j ; 8j 2 P > > > > i2N > > : jPj x 2 RjAj þ y 2 Y # f0; 1g
ð3:1Þ
In the above model the objective function minimizes total costs made of: the costs associated with opening and operating the facilities and the operational or transporting costs. The first constraint set enforces the flow conservation of product across any node j 2 P. The second constraint set ensures that the total flow of product to customer j satisfies the demand d j at least. The third constraint set guarantees that the total flow of product from supplier j is less that the supply d j . Finally, the last constraint set enforces the non-negativity restrictions on the corresponding decision variables and the integrality restrictions on the binary variables. It also includes the set Y and X of logical dependencies and restrictions. For simplicity, we have the following compact notation for model (3.1): min
s:t:
cT y þ t T x 8 Nx ¼ 0 > > > > > > < Dx 6 d Sx P s > > > Rx 6 my > > > : jPj x 2 RjAj þ ; y 2 Y # f0; 1g
ð3:2Þ
where vectors c, t, d, and s are corresponding to opening and operational costs, transporting costs, demands, supplies, respectively. The matrices N ; D, and S are appropriate matrices corresponding to the summations on the left-hand-side of the expressions (3.1), respectively. R corresponds to a matrix of rj , and m corresponds to a matrix of mj . To extend the above model to a random fuzzy setting, we assume that transportation costs, demands, and ^ ^sÞ represents the random fuzzy vector. The resultsupplies are random fuzzy variables. In particular, n ¼ ð^t; d; ing formulation is as followed:
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min s:t:
cT y þ ^tT x 8 ^ > < Dx 6 d Sx P ^s > : jPj ðx; yÞ 2 G0 ¼ fx 2 RjAj þ ; y 2 Y # f0; 1g jNx ¼ 0; Rx 6 myg
ð3:3Þ
which is a random fuzzy programming. 4. Solution method Considering problem (3.3). In practice, it would certainly be more appropriate to consider that the possible values of the fuzzy parameters in the probability distributions usually involve the ambiguity of the experts’s understanding of the real system. For this reason, in this paper, we consider the following auxiliary programming problem involving random fuzzy variables: min cT y þ E gTt x 8 PfDj x P gd j g P bd j ; 8j 2 C > > > < PfS x 6 g g P b ; 8i 2 S ð4:1Þ i si sj s:t: > ðx; yÞ 2 G0 > > : gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ with 0 6 bd j 6 1, 0 6 bsi 6 1, and Dj and S i indicating the j-th and i-th of D and S, respectively. Note that the first and second constraint sets are disjoint chance constraints. The above model is an (a; b) chance constrained programming. It should be emphasized here that the parameters gn ¼ ðgt ; gd ; gs Þ are treated as decision variables rather than constants. On the basis of the a-cut sets of the random fuzzy variables, we introduce the concept of a-optimal solutions to it: ¼ fxjPfDj x P gd j g P bd j ; 8j 2 C; g, G2 ðgs Þ ¼ Definition 4.1. (a-optimal solution) Set T G1 ðgd Þ T fxjPfS i x 6 gsi g P bsi ; 8i 2 S; g and G1 ðgd gs Þ ¼ G0 G1 ðgd Þ G2 ðgs Þ. A feasibility solution ðx ; y Þ of (4.1) is T T said to be an a-optimal solution, if and only if there exists cT y þ gT t x 6 c y þ gt x for any ðx; yÞ 2 Gðgd ; gs Þ; gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ where the corresponding values of parameters g ¼ ðgt ; gd ; gs Þ are called a-level optimal parameters. 4.1. The existence of solutions Let we consider the following problem: min cT y þ E gTt x 8 > < PfDj x P gd j g P bd j ; 8j 2 C PfS i x 6 gsi g P bsj ; 8i 2 S s:t: > : ðx; yÞ 2 G0 where gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ. According to Theorem 2.2, problem (4.2) is equivalent to min cT y þ E gTt x 8 Dj x P minfK b jPfK b P gd j g P bd j g; 8j 2 C > Kb > < S i x 6 maxfK b jPfK b P gsi g P bsi g; 8i 2 S s:t: Kb > > : ðx; yÞ 2 G0
ð4:2Þ
ð4:3Þ
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We now show that the feasibility set Gðgd ; gs Þ is bounded and closed. Assume that G0 , G1 ðgd Þ, G2 ðgs Þ–0. jPj According to x 2 RjAj þ ; y 2 Y # f0; 1g , Gðgd ; gs Þ is bounded. According to Proposition 2.1, Gðgd ; gs Þ is closed. That is the feasibility set Gðgd ; gs Þ is compact. Proposition 4.1. Let f be a function from Gðgd ; gs Þ to R and f ¼ cT y þ E gTt x x. We set f; if ðx; yÞ 2 Gðgd ; gs Þ fGðgd ;gs Þ ðx; yÞ :¼ þ1; if ðx; yÞ 2 ðX ; Y Þ n Gðgd ; gs Þ jAj jPj where S ðX ; Y Þ ¼ Rþ Rþ and fGðgd ;gs Þ ðx; yÞ is no longer a real-valued function but a function from ðX ; Y Þ to R fþ1g. Then fGðgd ;gs Þ ðx; yÞ is lower semi-continuous. jPj Proof. Firstly, we consider f1 ¼ cT y. The domain of function f1 is Dom f 1 :¼ y 2 RjPj þ jf1 < þ1 ¼ f0; 1g . jPj We take k 2 R, then Sðf1 ; kÞ :¼ y 2 Rþ jfS1 6 k # Dom f 1 . Obviously, anarbitrary section Sðf1 ; kÞ is closed. fþ1g, is lower semi-continuous. Next, we consider f2 ¼ E gTt x Thus, f1 ¼ cT y, which is from RjPj þ Sto R 2.1, an arbitrary section which is from RþjAj to R fþ1g. According to Proposition S jAj T jf < þk is closed. It follows that f ¼ E g to R fþ1g, is lower x which is from R Sðf2 ; kÞ :¼ x 2 RjAj 2 2 t þ þ semi-continuous. Resulted from Proposition 2.2, fGðgd ;gs Þ ðx; yÞ is lower semi-continuous. h
apply Theorem 2.1 to the function fGðgd ;gs Þ ðx; yÞ defined by fGðgd ;gs Þ ðx; yÞ ¼ cT yþ We jPj E gTt x if ðx; yÞ 2 Gðgd ; gs Þ and fGðgd ;gs Þ ðx; yÞ ¼ þ1; if x 2 ðX ; Y Þ n Gðgd ; gs Þ where ðX ; Y Þ ¼ RjAj þ Rþ , noting that fGðgd ;gs Þ ðx; yÞ is lower semi-continuous (see Proposition 4.1), that fGðgd ;gs Þ ðx; yÞ is lower semi-compact, and Gðgd ; gs Þ is relative compact. Solutions of problem (4.3) and so (4.2) exist. As is well known, a-cuts of each fuzzy number are closed intervals of real numbers for all a 2 ð0; 1, which is an important property of the fuzzy number. Obviously, the optimal solutions of problem (4.1) exist. 4.2. Deterministic equivalent In the paper, the demands and capacities of suppliers, have lognormal distributions with known scale parameters and unknown location parameters. Consider the case of the demand d j , having a lognormal distribution of which the mean value M d j is a triangular fuzzy number and the standard deviation were chosen as certain fractions of the mean value. Similarly, the capacity of supplier si , has a lognormal distribution of which the mean value M si is a triangular fuzzy number and the standard deviation were chosen as certain fractions of the mean value. Now from the properties of the a-level sets for the vectors of the fuzzy numbers M d j and hM si , it should i be noted that the a-level sets for M d j and M si can be denoted, respectively by the intervals M Ldj a ; M Rdj a and h i L M si a ; M Rsi a . The corresponding random variable intervals are gLdj a ; gRdj a and gLsi a ; gRsi a . For a given bd j , if M 1d j a 6 M 2d j a , k 1bd 6 k 2bd ; For a given bsi , if M 1si a 6 M 2si a , k 1bs 6 k 2bs . According to Theorem 2.1 the first and secj
j
i
i
ond constraint set of (4.3) are converted to Dj x P k bd j , S i x 6 k bsi , respectively, where Pfk bd j P gd j g ¼ bd j and Pfk bsi P gsi g ¼ bsi . Now we introduce the following set-valued functions. T dj ðk bd j Þ ¼ fxjd j x P k bd j g; T si ðk bsi Þ ¼ fxjS i x 6 k bsi g;
8j 2 C 8i 2 S
ð4:4Þ ð4:5Þ
Then, it can be verified that the following relations hold for T dj ðÞ and T si ðÞ, when x P 0. If k 1bd 6 k 2bd , then j j T dj ðk 1bd Þ T dj ðk 2bd Þ; If k 1bs 6 k 2bs , then T si ðk 1bs Þ T dj ðk 2bs Þ. Consequently, we can obtain an optimal solution to j j i i i i (4.1) by solving the following problem.
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min s:t:
T cT y þ M Tt 8 D x P k Lbd > > j < j R S x P k i b si > > : ðx; yÞ 2 G0
ð4:6Þ
where Pfk Lbd P gLdj g ¼ bd j and Pfk Rbs P gRsi g ¼ bsi . j i Then it is interesting to note that M Lt , k Lbd and k Rbs are a-level optimal parameters for any a-optimal solution, j i The relations between the optimal solutions to problem (4.6) and the a-optimal solution to (4.1) can be characterized by the following theorems. Theorem 4.1. If ðx ; y Þ is a optimal solutions to (4.6), then ðx ; y Þ is an a-optimal solution to the (4.1). Proof. Assume that ðx ; y Þ is not an a-optimal solution to the (4.1). Then, since M Lt , k Lbd and k Rbs are a-level j i optimal parameters to the (4.1), there exist ðx; yÞ 2 Gðgd ; gs Þ and gn ¼ ðgt ; gd ; gs Þ 2 La ðnÞ such that x . Then it holds that cT y þ E gTt x < cT y þ E gT t T T T LT cT y þ M LT t x 6 c y þ Mt x < c y Mt x
which contradicts the fact that ðx ; y Þ is a optimal solution to (4.6).
h
Theorem 4.2. If ðx ; y Þ is a optimal solution to (4.1), then ðx ; y Þ is an a-optimal solution to the (4.6). Proof. Assume that ðx ; y Þ is not an a-optimal solution to the (4.1). Since M Lt , k Lbd and k Rbs are a-level optimal j i parameters to the (4.1), according to Definition 4.1, ðx ; y Þ is a optimal solution to (4.6). h Obviously, problem (4.6) is the deterministic equivalent to the problem (4.1), when gd j ; 8j 2 C and gsj ; 8i 2 S follow lognormal distributions. At a given possibility level a, we can convert problem (4.1) to the formulation (4.6) which is a deterministic mixed-integer 0–1 programming. 5. st-GA Recently, there has been an increasing interest in using various evolutionary computation methods to solve hard optimization problems (Gen & Cheng, 1997). Among them, genetic algorithm (GA) is the one of the most well known class of evolutionary algorithms. The underlying principles of GAs were first exposed by Holland (1975). GAs deal with a coding of the problem instead of decision variables. They require no domain knowledge—only the payoff or objectives for evaluating fitness after operating genetic operations. In addition, traditional methods use deterministic transition rules to guide the search, such as hill-climbing, neighborhood search. Another difference between traditional methods and GAs is the latter searches from a set of points, while the former from a single point. These makes GAs more robust than traditional methods regarding their potential as optimization techniques to solve many real world problems (Gen & Cheng, 1997). Michalewicz (1994) was the first researcher, who used GA for solving linear and non-linear transportation/ distribution problems. In his method, he represented each chromosome of the problem by using m n matrix. The use of st-GA for solving some network problem was introduced and applied by Gen and Cheng (1997), and Syarif, Yun and Gen (2002). They employed Pru¨fer number in order to represent a candidate solution to the problems and developed feasibility criteria of Pru¨fer number to be decoded into a spanning-tree. The verification for the excellence of the Pru¨fer number was addressed by Zhou and Gen (1996). They noted that the use of Pru¨fer number is more suitable for encoding a spanning-tree, especially in some research fields, such as: transportation problems, minimum spanning-tree problems, and so on. Also, it is shown that we can use only m þ n 2 digit number to uniquely represent a distribution network with m suppliers and n destinations, where each digit is an integer between 1 and m þ n. This means that Pru¨fer number representation is more efficient in the sense of required memory for computation than that of matrix-based representation. In this paper, we consider the network problem (4.6). As the solution method, we use st-GA and design feasibility check and develop repairing procedure, so that it can be applied for relatively large size problems. The steps of our algorithm for solving the problem are listed as follows:
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Step 1. Set the initial values and the parameters of genetic algorithms: population size pop_size, crossover rate pc , and maximum generation maxgen. Step 2. Generate the initial population. Step 3. Generic operate: crossover and mutation. Step 4. Check the feasibility of the offspring and repair the infeasible offspring. Step 5. Evaluate and select the chromosomes. Step 6. Repeat the second to fifth steps for the given number of maxgen. Step 7. Report the best chromosome as the optimal solution.
5.1. Representation and feasibility of the chromosome For this problem, we use sub-tree S P , sub-tree P W , sub-tree W C to represent the distribution pattern for suppliers to plant, plant to warehouses, and warehouses to customers, respectively. Each chromosome consists of five parts. The first part is Pl binary digits to represent the opened/closed plants. The second part is Wh binary digits to represent opened/closed warehouses. The last three parts are three Pru¨fer numbers representing the distribution pattern of each stage, respectively. In the first step which generates the chromosome, we generate the two 0–1 variables and check the total capacities of open plants and open warehouses to satisfy all the customer demands. If the total capacity is less than the total demand, then open the closed facility with maximum capacity until the total capacity of facilities is enough to satisfy the customer demands. Generating the three Pru¨fer numbers is followed. For the sub-tree W C, denote the suppliers set O ¼ f1; 2; . . . ; W g, and plants set D ¼ fW þ 1; W þ 2; . . . ; W þ Cg. Obviously, this distribution has W þ C nodes, which means that we need W þ C 2 digits Pru¨fer number in the range ½1; W þ C to uniquely represent the sub-tree W þ C. 5.1.1. Feasibility check for Pru¨fer number In the Pru¨fer number P ðT Þ, let P ðT Þ be the set of all nodes not included in P ðT Þ. Ri denotes the number of appearances of node i 2 P ðT Þ and Li denotes the number of connections of node i 2 O [ D. The Pru¨fer number P ðT Þ is said to be feasible if X X Li ¼ Li ð5:1Þ i2O
i2D
The feasibility check and repairing procedure for the Pru¨fer number is as follows: Step 1. Compute Ri for i 2 O [ D in P ðT Þ Step 2. Li P ¼ Ri þ 1 P Step 3. If i2O Li > i2D Li , then select one digit in P ðT Þ which contains node i 2 O and replace it with the number j 2 D generated randomly. Otherwise, select one digit in P ðT Þ which contains node i 2 D and replace it with the number j 2 O generated randomly. After checking for the feasibility, the Pru¨fer number can be decoded into spanning-trees in order to determine the distributaries pattern for the network. The procedures of decoding Pru¨fer number is given as follows: Step 1. Let i be the smallest label node of P ðT Þ and j be the leftmost digit of P ðT Þ. Step 2. If i and j are not in the same set O or D, add the edge from i to j into the tree. Otherwise, select the next digit k from P ðT Þ that not included in the same set with i. Exchange j with k, add the edge ði; kÞ to the tree. Step 3. Remove j(or k) form P ðT Þ and i from P ðT Þ. If j(or k) does not appear anywhere in the remaining part of P ðT Þ, put it into P ðT Þ. Designate i as no longer eligible. Step 4. Assign the available amount of units to xxj ¼ minfai ; bj g or xij ¼ minfai ; bk g. Step 5. Update the availability ai ¼ ai xij and bj ¼ bj xij Step 6. Repeat the process Step 1–Step 5, until no digits left in P ðT Þ.
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Step 7. If no digits remain in P ðT Þ, there are exactly two nodes, r and s, still negligible for consideration. Add link from r to s into the tree and form a tree with W þ C 1 links. Step 8. If no available amount of units to assign, then stop. Otherwise, there are remaining supply of node r and demand of node s, add edge (r; s) to tree and assign the available amount of units xrs ¼ minfar ; bs g to edge. If there exists a cycle, then remove the edge that assigned zero flow. By using similar ways, we can generate two other Pru¨fer numbers representing P W and S P sub-trees. These two Pru¨fer numbers have P þ W 2 and S þ P 2 digits, respectively. Note that in order to ensure the flow conservation of product across any node, we first decode the Pru¨fer number W C, then P W and finally S P . When decoding the chromosome, firstly update the capacity of open facilities by their total exporting amount to the next stage and change the capacity of the closed facilities to be zero. 5.2. Genetic operations The genetic algorithm uses the individuals in the current generation to create the children that make up the next generation. Besides elite children, which correspond to the individuals in the current generation with the best fitness values, the algorithm creates crossover children by selecting vector entries, or genes, from a pair of individuals in the current generation and combines them to form a child and mutation children by applying random changes to a single individual in the current generation to create a child. Both processes are essential to the genetic algorithm. Crossover enables the algorithm to extract the best genes from different individuals and recombine them into potentially superior children. Mutation adds to the diversity of a population and thereby increases the likelihood that the algorithm will generate individuals with better fitness values. Without mutation, the algorithm could only produce individuals whose genes were a subset of the combined genes in the initial population. 5.2.1. Crossover The crossover is done by exchanging the information of two parents to provide a powerful exploration capability. In this paper, we employ a one-cut-point and exchanges the right parts of two parents to generate offspring. However, before decoding all Pru¨fer numbers, we still need to check whether the capacity of the opened facilities is enough to satisfy the demand or not, and check the feasibility of the Pru¨fer numbers. Fig. 2 shows the one-point crossover process. 5.2.2. Mutation Mutation adds to the diversity of a population and thereby increases the likelihood that the algorithm will generate individual with better fitness values. Without mutation, the algorithm could only produce individuals whose genes were a subset of the combined genes in the initial population. We use inversion mutation operations. The inversion mutation selects two positions within a chromosome at random and then inverts the sub-string between these two positions. This mutation operators will generate
Fig. 2. The one-point crossover process.
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Fig. 3. The inversion mutation process.
feasible Pru¨fer numbers if the parents are feasible. However, we also need to check the feasibility of the opened facilities to satisfy the demand. Fig. 3 shows the illustration of those mutation processes. We can specify how many of each type of children the algorithm creates as follows: (a) Elite count, in reproduction options, specifies the number of elite children. (b) Crossover fraction, in reproduction options, specifies the fraction of the population, other than elite children, that are crossover children. For example, if the population size is 20, the elite count is 2, and the crossover fraction is 0.8, the numbers of each type of children in the next generation are as follows: There are 2 elite children and 18 individuals, so the algorithm rounds 0.8 18 = 14.4 to 14 to get the number of crossover children. The remaining 4 individuals, other than elite children, are mutation children. 6. Computational results In this section we describe numerical example using the proposed methodology and st-GA for solving a realistic multi-stage SC design problems, then demonstrate the computational effectiveness afforded by our method. Our test problem is that of a domestic supply chain network for a company. Henceforth this problem is referred to as the ‘‘domestic” problem. The supply chain topology with all possible center locations and transportation channels for the domestic problem is presented in Fig. 4. The main characteristics of the networks are presented in Table 1. We develop fuzzy mean values for the random fuzzy parameters from the values for the operational or transportational cost, demand and supply capacity parameters from Shapiro (2001). The standard deviations 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 supplier plant warehouse customer
0.2 0.1
0
0.2
0.4
0.6
0.8
1
1.2
Fig. 4. The distribution graph of hospitals and clinics.
1.4
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Table 1 Supply chain network characteristics.
Total facilities Supplies Plants Warehouses Customers Transportation channels
Domestic
Size of the problem
27 3 10 17 230 4110
Constraints Binary variables Total variables Coefficients
288 27 4137 12584
Table 2 Distribution pattern of the first and second stage. Suppliers (capacity)
Plants 1
2
3
4
5
6
7
8
9
10
1(14735) 2(13237) 3(14543)
4579 – –
– 2893 –
– 3707 –
– 3515 –
– – 3775.8
– – –
– – 3387
– 2451 –
– – –
– – 2444.3
Plant capacity Plants
4579 2893 Warehouses
3707
3515
4563
4172
3387
2451
4245
3192
1 2 3 4 5 6 7 8 9 10 Warehouse Capacity
1
2
3
4
5
6
7
8
9
10
11
– – 2426.1 – – – – – – 2144.5 – – – – – – – – – –
– – – – – – – – – – – – – – – – – – – –
– – – – – 1466.3 – – – – – – 2730 – – – – – – –
261.7 2255.9 – – – – – – – – – – – – 2451 – – – – –
– 891.3 – – – – – – – – – – – – – – – – – –
– – – – – 864.6 – 998.3 – – – – – – – – – – 2444.3 –
–
1170
–
–
–
–
466.9
–
–
–
–
–
1376.1
–
–
–
2516.7
–
–
–
– 29.2 –
750.2
–
851.9
–
–
–
–
–
–
–
–
657
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
2426.1 2519.7
2351.1 2586.3
2730 2566.6
2712.7 2255.9
2708.6 2525.4
2444.3 2614.9
2516.7
2387.1
2725.6
2350.3
2390.6
for the distributions were chosen as certain fractions of the mean values. The linear and mixed-integer programming problem (4.6) was solved by st-GA, which is implemented using Matlab7.0 program language. All computations were carried out on a Pentium IV 2.4GHZ PC running Windows XP with GA parameters pop-size = 100, P c ¼ 0:8 and elite count = 2. One heuristic distribution pattern solution of the first and second stage of this case study is shown in Table 2. It is shown that our methodology and st-GA is effective. However, since the search apace of this kind of problem is so large, it is very important to set the experiment with reasonable population size and maximum generation in order to ensure that we have a good results. More alternatives can be generated by varying the values of a and b.
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7. Conclusions Multifold uncertain programming is a growing subject. Here we provided some further researches in the random fuzzy programming area, which is an important branch of multifold uncertain programming. For the mathematical properties, we considered the methodology of a class of random fuzzy programming model, deterministic equivalent, and solution existence. Other aspects, such as sensitivity analysis, dual theorems are waited to investigated. From the computational viewpoint, we developed a st-GA for the deterministic equivalent model which is also used for solving other mixed 0–1 integer mathematic model of this class. From the applied viewpoint, we considered the random fuzzy environment in multi-stage SC design and applied our methodology to the random fuzzy mathematic model presented. Random fuzzy programming may be applied to any optimization problems with random fuzzy factors, for example, queuing system, finance, energy system. Acknowledgements This research was supported by the National Science Fund for Distinguished Young Scholars, PR China (Grant No. 70425005), and the Teaching and Research Award Program for Outstanding Young Teacher in Higher Eduction Institutions of MOE of PR China (Grant No. 20023834-3). References Aikens, C. H. (1985). Facility location models for distributing planning. European Journal of Operational Research, 22(3), 263–279. Amiri, A. (2004). Designing a distribution network in a supply chain system: Formulation and efficient solution procedure. European Journal of Operational Research. Aubin, J. P. (1993). Optima and equilibria. Berlein Heidelberg: Springer-Verlag. Charnes, A., & Cooper, W. W. (1959). Chance-constrained programming. Management Science, 6(1), 73–79. Cheung, R. K.-M., & Powell, B. W. (1996). Models and algorithms for distribution problems with uncertain demands. Transportation Science, 30, 822–844. Gen, M., & Cheng, R. (1997). Genetic algorithms and engineering design. New York: Wiley. Geoffrion, A. M., & Graves, G. W. (1974). Multi-commodity distribution system design by benders decomposition. Management Science, 20, 822–844. Geoffrion, A. M., & Powers, R. F. (1995). Twenty years of strategic distribution system design: An evolution perspective. Interfaces, 25, 105–128. Goetschalckx, M., Vidal, C. J., & Dogan, K. (2002). Modeling and design of global logistics systems: A review of integrated strategic and tactical models and design algorithms. European Journal of Operational Research, 143, 1–18. Guille´n, G., Mele, F. D., Bagajewicz, M. J., Espun˜a, A., & Puigjaner, L. (2005). Multiobjective supply chain design under uncertainty. Chemical Engineering Science, 60, 1535–1553. Holland, J. (1975). Adaptation in natural and artificial systems. Ann Arbor: University of Michigan Press. Landeghen, H. V., & Vanmaele, H. (2002). Robust planning: A new paradigm for demand chain planning. Journal of Operations Management, 20, 769–-783. Liu, B. (2002). Theory and practice of uncertain programming. Physica-berlag. Michalewicz, Z. (1994). Genetic algorithms + data structures = evolution programs (2nd ed.). New York: Springer. Min, H., & Zhou, G. (2002). Supply chain modeling: Past, present and future. Computers and Industrial Engineering, 43, 231–249. Nattier, W. (2001). Random fuzzy variables of second order and applications to statistical inference. Information Sciences, 133, 69–88. Owen, S. H., & Daskin, M. S. (1998). Strategic facility location: A review. European Journal of Operational Research, 111, 423–447. Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. European Journal of Operational Research, 167, 96–115. Shapiro, J. F. (2001). Modeling the supply chain. Thomson Learning. Syarif, A., Yun, Y. S., & Gen, M. (2002). Study on multi-stage logistic chain network: A spanning tree-based genetic algorithm approach. Computers & Industrial Engineering, 43, 299–314. Yan, H., Yu, Z., & Cheng, T. C. E. (2003). A strategic model for supply chain design with logical constraints: Formulation and solution. Computers & Operations Research, 30, 2135–2155. Zhou, G., & Gen, M. (1996). A note on genetic algorithms approach to the degree-constrained spanning tree problem. Networks, 30, 105–109. Jiuping Xu was born in Chongqing, China, in 1962. He received the Ph.D. of applied mathematics from Tsinghua University, Beijing, China; and the Ph.D. of physical chemistry from Sichuan University, Chengdu, China, in 1995 and 1999, respectively. He is currently Professor at Sichuan University and the vice-president of The Systems Engineering Society of China. His current research interests are in the areas of systems science and engineering, applied mathematics, and physical chemistry, management science and engineering man-
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agement. He has published 31 books and over 300 journal papers in IEEE Transaction on Fuzzy Systems, Fuzzy Sets and Systems, The International Journal of Management Sciences, Information Sciences, Journal of Computational and Applied Mathematics, Mathematics and Computers in Simulation, and Mathematical and Computer Modelling. Dr. Xu was received the IFORS Prize for Operational Research in Development of International Federation of Operations Research Societies at Vancouver, in 1996. He has received numerous prizes in China, including the China Youth Prize of Science and Technology, in 2004. Yanan He received M.S. degree from Sichuan University, Chengdu, China, in 2006. She is currently working toward the Ph.D. degree at Chinese Academy of Sciences, China. Her main research interests are in the fields of decision making, uncertain programming, and energy economics. Mitsuo Gen received the Ph.D. degree from the Kogakuin University, Japan in 1974. He was Lecturer at Ashikaga Institute of Technology, Japan in 1974–1980; Assoc. Professor at Ashikaga Institute of Technology, Japan in 1987.4–2003.3; He is currently a Professor of Graduate School of Information, Production & Systems, Waseda University. He was Visiting Assoc. Prof. at University of California at Berkeley, USA in 1999–2000. His research interests include Genetic Algorithms, Neural Network, Fuzzy Logic, and the applications of evolutionary techniques to Network Design, Schedule, System Reliability Design,etc. He has authored Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York (2000) with Dr. Runwei Cheng. Dr.Gen is a member of the editorial board of several international journals, such as, Computational Intelligence of Computers and Industrial Engineering, OR Spectrum, Fuzzy Optimization and Decision Making, International Journal of Manufacturing Technology & Management and International Journal of Smart Engineering & Systems Design.