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M. Jiménez, M. Parra, A. Bilbao-Terol, Maria Rodríguez (2007)
Linear programming with fuzzy parameters: An interactive method resolutionEur. J. Oper. Res., 177
Reay-Chen Wang, Tien-Fu Liang (2004)
Application of fuzzy multi-objective linear programming to aggregate production planningComput. Ind. Eng., 46
Yun-Kyong Kim, B. Ghil (1997)
Integrals of fuzzy-number-valued functionsFuzzy Sets Syst., 86
R. Bellman (1953)
Bottleneck Problems and Dynamic Programming.Proceedings of the National Academy of Sciences of the United States of America, 39 9
L. Fleischer, J. Sethuraman (2005)
Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and ExtensionsMath. Oper. Res., 30
B. Julien (1994)
An extension to possibilistic linear programmingFuzzy Sets and Systems, 64
M. Pullan (1993)
An algorithm for a class of continuous linear programsSiam Journal on Control and Optimization, 31
Xinwang Liu (2001)
Measuring the satisfaction of constraints in fuzzy linear programmingFuzzy Sets Syst., 122
J. Buckley (1988)
Possibilistic linear programming with triangular fuzzy numbersFuzzy Sets and Systems, 26
M. Pullan (1997)
Existence and duality theory for separated continuous linear programs, 3
David Peidro, J. Mula, M. Jiménez, Ma Botella (2010)
A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environmentEur. J. Oper. Res., 205
E. Anderson (1981)
A new continuous model for job-shop schedulingInternational Journal of Systems Science, 12
Congxin Wu, Z. Gong (2001)
On Henstock integral of fuzzy-number-valued functions (I)Fuzzy Sets Syst., 120
Xuzhu Wang, E. Kerre (2001)
Reasonable properties for the ordering of fuzzy quantities (II)Fuzzy Sets Syst., 118
R. Buie, J. Abrham (1973)
Numerical solutions to continuous linear programming problemsZeitschrift für Operations Research, 17
J. Buckley (1989)
Solving possibilistic linear programmingFuzzy Sets and Systems, 31
Abbas ZIVARI-KAZEMPOUR, Mohammad Hadadi (1992)
Mathematical analysis
H. Rommelfanger (1996)
Fuzzy linear programming and applicationsEuropean Journal of Operational Research, 92
A. Philpott, M. Craddock (1995)
An adaptive discretization algorithm for a class of continuous network programsNetworks, 26
R. Słowiński, J. Teghem (1990)
Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty
P. Vasant (2003)
Application of Fuzzy Linear Programming in Production PlanningFuzzy Optimization and Decision Making, 2
Y. Nazarathy, Gideon Weiss (2009)
Near optimal control of queueing networks over a finite time horizonAnnals of Operations Research, 170
E. Anderson, A. Philpott (1994)
On the Solutions of a Class of Continuous Linear ProgramsSiam Journal on Control and Optimization, 32
R. Goetschel, W. Voxman (1986)
Elementary fuzzy calculusFuzzy Sets and Systems, 18
H. Maleki, M. Tata, M. Mashinchi (2000)
Linear programming with fuzzy variablesFuzzy Sets Syst., 109
N. Mahdavi-Amiri, S. Nasseri (2007)
Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variablesFuzzy Sets Syst., 158
Shaocheng Tong (1994)
Interval number and fuzzy number linear programmingsFuzzy Sets and Systems, 66
Young-Jou Lai, C. Hwang (1992)
A new approach to some possibilistic linear programming problemsFuzzy Sets and Systems, 49
H. Zimmermann (1985)
Fuzzy Set Theory - and Its Applications
M. Roubens (1990)
Inequality Constraints between Fuzzy Numbers and Their Use in Mathematical Programming
M. Pullan (1996)
A Duality Theory for Separated Continuous Linear ProgramsSiam Journal on Control and Optimization, 34
Jolly Puri, Amit Kumar (2009)
Fuzzy Linear Programming and its Applications
B. Hajek, R. Ogier (1982)
Optimal dynamic routing in communication networks with continuous traffic1982 21st IEEE Conference on Decision and Control
Liang-Hsuan Chen, Wen-Chang Ko (2009)
Fuzzy linear programming models for new product design using QFD with FMEAApplied Mathematical Modelling, 33
Andreas Ernst (1996)
Continuous-time quadratic cost flow problems with applications to water distribution networksThe Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 37
Hideo Tanaka, P. Guo, H. Zimmermann (2000)
Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problemsFuzzy Sets Syst., 113
Chitrasen Samantra (2012)
Decision-making in fuzzy environment
H. Maleki (2003)
RANKING FUNCTIONS AND THEIR APPLICATIONS TO FUZZY LINEAR PROGRAMMING, 4
Xiaodong Luo (1995)
Continuous linear programming : theory, algorithms and applications
Congxin Wu, Z. Gong (2000)
On Henstock integrals of interval-valued functions and fuzzy-valued functionsFuzzy Sets Syst., 115
Paul Schrimpf (2011)
Dynamic Programming
(2009)
An approximation algorithm for separated continuous linear programs with fuzzy valued objective functions
E. Anderson, P. Nash, André Perold (1983)
Some Properties of a Class of Continuous Linear ProgramsSiam Journal on Control and Optimization, 21
M. Pullan (1995)
Forms of Optimal Solutions for Separated Continuous Linear ProgramsSiam Journal on Control and Optimization, 33
R. Yager (1981)
A procedure for ordering fuzzy subsets of the unit intervalInf. Sci., 24
Fuzzy Inf. Eng. (2010) 1: 5-26 DOI 10.1007/s12543-010-0034-9 O RIGINAL AR T IC L E Solution Algorithms for a Class of Continuous Linear Programs with Fuzzy Valued Objective Functions Mohammad Mehdi Nasrabadi· Mohammad Ali Yaghoobi· Mashaallah Mashinchi Received: 25 November 2009/ Revised: 20 January 2010/ Accepted: 16 February 2010/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2010 Abstract This paper discusses a class of continuous linear programs with fuzzy val- ued objective functions. A member of this class is called a fuzzy separated continuous linear program (FSCLP). Such problems have applications in a number of domains, including, production and inventory systems, communication networks, and pipeline systems for transportation. The discretization approach is used to construct two ordi- nary fuzzy linear programming problems, which give a lower and an upper bound on the optimal value of FSCLP. It is then shown how to construct an improved feasible solution for FSCLP starting from a nonoptimal one. This leads to the development of a class of algorithms based on a sequence of discrete approximations to FSCLP. Nu- merical examples in the context of continuous-time networks are presented to show the applicability of the proposed method. Keywords Continuous linear programming· Fuzzy linear programming· Duality· Discretization 1. Introduction Bellman [5,6] introduced a class of optimization problems to model some economical processes which he called “Bottleneck Processes”. These problems are generally referred to as continuous linear programs (CLP), since they can be formulated as Mohammad Mehdi Nasrabadi () Department of Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Ker- man, Kerman, Iran email: m m nasrabadi@yahoo.com − − Mohammad Ali Yaghoobi · Mashaallah Mashinchi Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran 6 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) linear programs with variables which are functions of time as follows: CLP: min c(t) x(t)dt s.t. B(t)x(t)+ K(s, t)x(s)ds ≤ b(t), x(t) ≥ 0, t ∈ [0, T ]. Here T > 0 is a given time horizon, and for each t ∈ [0, T ], B(t), K(t, s), s ∈ [0, T ], are given n × n matrices, c(t)isa given n -vector, b(t) isagiven n -vector and x(t) 2 1 1 2 is an n -vector to be determined. All vectors are as columns and the superscript “ ” denotes the transpose operation. Anderson [1] introduced a class of continuous linear programs called separated continuous linear programs (SCLP) in order to model job-shop scheduling problems. SCLP takes the following form: SCLP: min c(t) x(t)dt s.t. Gx(s)ds+ y(t) = a(t), Hx(t) ≤ b(t), x(t) ≥ 0, y(t) ≥ 0, t ∈ [0, T ], where G and H are given matrices of dimensions n × n and n × n , and c(t), a(t) 2 1 3 1 and b(t) are given vectors of dimensions n , n and n , respectively, as functions of 1 2 3 time t ∈ [0, T ]. The decision variables are given by x(t) = (x (t),..., x (t)) and 1 n y(t) = (y (t),..., y (t)) , which are vector valued functions of dimension n and n . 1 n 1 2 Moreover, x(t), c(t) and b(t) are bounded measurable functions, and y(t) and a(t) are absolutely continuous functions. Since the introduction of SCLP, a number of authors have become interested in developing conditions under which an optimum solution exists with a finite number of breakpoints. In particular, Anderson et al [2] characterized the extreme point solu- tions of SCLP and showed that an optimal solution with a finite number of breakpoints in certain cases exists. Anderson and Philpott [3] discussed the form of solutions for SCLP. They proved the existence of a piecewise analytic optimum solution under cer- tain assumptions and established a strong duality result. Shortly after Anderson and Philpott [3], Pullan [30] showed under various different assumptions that there exist extreme point optimal solutions for SCLP which are piecewise constant, piecewise polynomial, or, more generally, piecewise analytic. In 1996, Pullan [31] developed a detailed duality theory for a class of SCLP, based on a particular dual problem. He introduced a notion of complementary slackness using a weak duality result and developed several sufficient optimality conditions for SCLP. Moreover, Pullan established a fairly general condition for the absence of a duality gap and proved a strong duality result between SCLP and its dual when the input functions (i.e., c(t), a(t), and b(t)) are piecewise analytic. The development of algorithms to solve any form of SCLP has attracted the atten- tion of many researchers. The standard approach is to solve the original continuous Fuzzy Inf. Eng. (2010) 1: 5-26 7 linear program via a series of discrete approximations by partitioning the time interval [0, T ] into finitely many subintervals. This approach, which is called discretization, was first taken by Buie and Abraham [10] for solving CLP and later improved by Pullan [29] and Philpott and Craddock [28] for SCLP with piecewise constant/linear input functions. The discretization approach has attracted most of the attention for solving practical problems because the solutions for the discrete approximations con- verge to the solution for SCLP as the partitions become finer. The SCLP problem arises in a number of application areas such as production and inventory systems [13, 17], communication networks [15, 17], telephone loss net- works [17], fluid queueing networks [17, 26], and pipeline systems for transportation [12, 32]. A crucial feature in such applications is that all or some of parameters are imprecise and ambiguous. This characteristic is not captured by SCLP at all and in- put data must be well defined and precise. In engineering applications, this is not a realistic assumption and usually the value of many parameters of a model is estimated by experts. Clearly, it cannot be assumed the knowledge of experts is so precise. One practical way is to express the uncertain parameters by fuzzy numbers. Although again in this approach the knowledge of experts may be utilized, the parameters are not expressed by deterministic data. They are estimated in terms of fuzzy numbers which are more realistic and create a conceptual and theoretical framework for deal- ing with imprecision and vagueness [16, 36]. During the last few decades, significant progress has been made in solving fuzzy linear programs since Bellman and Zadeh [7] proposed the notion of fuzzy decision making [8, 9, 11, 16, 18, 20, 21, 23, 27, 33-36, 38, 40]. In contrast, to best of our knowledge, no attempt has been made in solving fuzzy linear programs in infinite- dimensional spaces despite their importance as models in engineering applications. As a starting point in this direction, the authors of the present work introduced the following class of separated continuous linear programs with fuzzy valued objective functions (hereafter called FSCLP): FSCLP: min (˜γ+ ct ˜ ) x(t)dt s.t. Gx(s)ds+ y(t) = α+ at, t ∈ [0, T ], (1) Hx(t) ≤ b, t ∈ [0, T ], (2) x(t) ≥ 0, y(t) ≥ 0, t ∈ [0, T ]. (3) n n n n 1 2 3 1 In contrast to SCLP, hereγ, ˜ c ˜ ∈ FN(R) ,α, a ∈ R , b ∈ R , where FN(R) denote the set of n -vectors whose components are fuzzy numbers. Nasrabadi et al [25] constructed two different discretizations of FSCLP by parti- tioning the time interval [0, T ] into finitely many subintervals. One discretization is based on averaging the properties of the problem over each subinterval which gives an upper bound on the optimal value of FSCLP. Another one corresponds to a dual problem FSCLP which gives a lower bound on the optimal value of FSCLP. Both of these discretized problems can be solved by any of the methods for solving fuzzy linear programming problems. They [25] showed that the gap between lower and 8 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) upper bounds approaches to zero when the time discretization gets arbitrarily fine, and derived a strong duality result. Moreover, they developed an algorithm, so-called uniform discretization, for solving FSCLP based on dividing the time interval into a series of subintervals of equal length. In this paper, following our previous work [25], we present another algorithm based on a step to construct an improved feasible solution to the problem starting from a nonoptimal one. This algorithm solves the two discretized problems on suc- cessively finer (nonuniform) discretizations. In contrast to the uniform discretization algorithm presented in [25], the algorithm uses the properties of the discretized prob- lems to insert new breakpoints at appropriate places in order to improve the current solution. The rest of this paper is structured as follows. Section 2 provides preliminaries from the fuzzy set theory needed for the purposes of this paper. We present two dis- crete approximation problems of FSCLP followed by a discussion of their properties in Section 3. We extend the algorithm proposed by Pullan [29] for SCLP to FSCLP in a fuzzy environment in Section 4. We illustrate the developed algorithm by small examples in Section 5. Finally, Section 6 is devoted to conclusion. 2. Preliminaries Here we give some definitions for the treatment of FSCLP. First, for the convenience of the reader, we introduce fuzzy numbers and some of the well-known notations and definitions. Let F(R) denote the set of all fuzzy subsets of the real line R. We say that a ˜ ∈ F(R) is a fuzzy number if it has the following properties: 1) a ˜ is normal fuzzy set, i.e., there exists an x ∈ R with a ˜ (x ) = 1; 0 0 2) a ˜ is convex fuzzy set, i.e., a ˜ (λx+ (1−λ)y) ≥ min{a ˜ (x), a ˜ (y)} for any x, y ∈ R,λ ∈ [0, 1]; 3) a ˜ is upper semicontinuous; 4) supp(a ˜):= cl{x ∈ R :˜ a(x) > 0} is compact, where cl(.) denotes closure of a set. We use the notation FN(R) to denote the set of all fuzzy numbers. For any h ∈ [0, 1], the h-level set of a fuzzy set a ˜ is defined by x :˜ a(x) ≥ r, 0 < h ≤ 1, [˜ a] = h ⎪ supp(a ˜ ), h = 0. It is easy to establish that a ˜ is a fuzzy number if and only if (see [43]) 1) [a ˜ ] is a closed and bounded interval for each h ∈ [0, 1]; 2) [a ˜ ] is nonempty. From this characterization of fuzzy numbers, it follows that a fuzzy number a ˜ is L R completely determined by the end points of the intervals [a ˜ ] = [˜ a , a ˜ ]. h h Fuzzy Inf. Eng. (2010) 1: 5-26 9 Theorem 1 (Goetschel and Voxman [14]). For each a ˜ ∈ FN(R) and each h ∈ [0, 1], L L R R let a ˜ (h):= a ˜ and a ˜ (h):= a ˜ denote the left and right endpoints, respectively, of h h the h-level interval of a ˜ . Then i) a ˜ (h) is a bounded increasing function on [0, 1]; ii) a ˜ (h) is a bounded decreasing function on [0, 1]; L R iii) a ˜ (1) ≤ a ˜ (1); L R iv) a ˜ (h) and a ˜ (h) are left continuous on (0, 1] and right continuous at 0; L R v) If a ˜ (h) and a ˜ (h) satisfy (1)-(4) above, then there exists a unique b ∈ FN(R) L L R R ˜ ˜ such that b = a ˜ (h) and b = a ˜ (h). h h By Theorem 1, we can identify a fuzzy number a ˜ with the parameterized represen- tation L R {(˜ a , a ˜ ): 0 ≤ h ≤ 1}. h h Let a ˜, b ∈ FN(R),λ ∈ R. Based on extension principle [43], the addition and scalar multiplication are defined by ˜ ˜ a ˜ + b (z):= sup min{a ˜ (x), b(y)}; x+y=z a ˜ (z/λ), if λ 0; (λa ˜ )(z):= 0, if λ = 0. It easily follows that L L R R ˜ ˜ ˜ [˜ a+ b] = [˜ a + b , a ˜ + b ], h h h h ⎪ L R [λa ˜ ,λa ˜ ], ifλ ≥ 0; h h [λa ˜ ] = ⎪ R L λa ˜ ,λa ˜ , ifλ< 0. h h Orderings and rankings of fuzzy numbers play a fundamental role in optimization and decision-making problems. Throughout the past years, different approaches to this issue have been introduced (see [39] and references therein). An effective ap- proach is to define a ranking function R : FN(R) → R, where every fuzzy number is mapped into a point on the real line, where a natural order exists. Then, a ranking on fuzzy numbers can be defined as: ˜ ˜ � a ˜ ≥ b if and only if R(˜ a) ≥ R( b); ˜ ˜ � a ˜ = b if and only if R(˜ a) = R( b); ˜ ˜ � a ˜ > b if and only if R(˜ a) > R( b). ˜ ˜ ˜ where a ˜ and b are two fuzzy numbers. Also a ˜ ≤ b if and only if b ≥ a ˜ . R R ˜ ˜ Remark 1 We use the notation a ˜ = b without the subscript R when a ˜ and b have the same membership functions, and the notation a ˜ ≥ 0 when a ˜ (t) = 0 for every t < 0. 10 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) A ranking function R is said to be linear if ˜ ˜ R a ˜ + kb = R (a ˜ )+ kR b for any a ˜, b ∈ FN(R) and any k ∈ R. The linear ranking functions have been mostly used in solving fuzzy linear pro- gramming (see[22-24]). In this paper, we restrict our attention to linear ranking func- tions. In fact, the results that we present later are valid for any arbitrary, but fixed, linear ranking function R. There are many ranking functions which have been de- fined by authors according to their requirements (see [22, 39]). Some examples are as follows: 1) Yager [37] proposed a ranking function based on the concept of h-level sets. L R Let a ˜ = [˜ a , a ˜ ] be the h-level set of a ˜ . Then, the ranking function proposed h h by Yager [37] is defined as L R R(˜ a) = (˜ a + a ˜ ) dh. h h 2) Roubens [34] introduced a ranking function based on the compensation of areas determined by the membership functions. Let a ˜ and b be fuzzy numbers and L R ˜ ˜ S (˜ a, b) and S (˜ a, b) be the areas determined by their membership functions according to the following formulas: L L L ˜ ˜ S (˜ a, b) = a ˜ − b dh, h h I(˜ a,b) L L ˜ ˜ where I(˜ a, b) = h :˜ a ≥ b , is a subset of [, 1],> 0 and h h R R R ˜ ˜ S (˜ a, b) = a ˜ − b dh, h h S (˜ a,b) R R ˜ ˜ where S (˜ a, b) = h :˜ a ≥ b , is a subset of [1,],> 0. h h According to Roubens method [34], the degree to which a ˜ ≥ b is defined as L L R R ˜ ˜ ˜ ˜ ˜ C(˜ a, b) = S (˜ a, b)− S (b, a ˜ )+ S (˜ a, b)− S (b, a ˜ ). Then a ranking on fuzzy numbers can be defined as follows: ˜ ˜ � a ˜ ≥ b if and only if C(˜ a, b) ≥ 0; ˜ ˜ � a ˜ = b if and only if C(˜ a, b) = 0; ˜ ˜ � a ˜ > b if and only if C(b, a ˜ ) ≥ 0. Among fuzzy numbers, trapezoidal fuzzy numbers (TFNs) are of the greatest im- portance due to their applications. A TFN a ˜ can be represented by L L R R a ˜ = {(a − (1− h)α , a + (1− h)α ): 0 ≤ h ≤ 1}, Fuzzy Inf. Eng. (2010) 1: 5-26 11 L R L R in which a and a are called the left and right centers, respectively, andα andα are L R L R called the left and right spreads, respectively. We use the notation a ˜ = (a , a ,α ,α ) to represent a TFN and TFN(R) to denote the set of all TFNs. The following result is easily established (see [43]). L R L R L R L R Lemma 1 Let a ˜ = (a , a ,α ,α ) and b = (b , b ,β ,β ) be two TFNs. Then we have L L R R L L R R Addition: a ˜ + b = (a + b , a + b ,α +β ,α +β ). ⎪ L R L R (λa ,λa ,λα ,λα ), if λ ≥ 0; Scalar multiplication: λa ˜ = ⎪ L R R L (λa ,λa ,−λα ,−λα ), ifλ< 0. The Lebesgue integral of fuzzy number valued functions have been discussed by a number of authors (see [19, 41, 42] and the references therein). Here we define Lebesgue integral of fuzzy number valued functions slightly different from those in the mentioned works by using Theorem 1. Let f :[a, b] → FN(R) be a fuzzy number valued function and L R ˜ ˜ ˜ [ f (t)] = [ f (t), f (t)], t ∈ [a, b], h ∈ [0, 1]. h h L R ˜ ˜ ˜ We say that f is Lebesgue-integrable on [a, b] if the functions f and f are both h h Lebesgue-integrable on [a, b], for any h ∈ (0, 1]. Suppose that f :[a, b] → FN(R)is Lebesgue-integrable. The Lebesgue integral of f over [a, b] is defined to be a fuzzy set as b b b L R ˜ ˜ ˜ f (t) dt (x):= sup h ∈ [0, 1] : x ∈ f (t) dt, f (t) dt , x ∈ R. h h a a a ˜ ˜ Lemma 2 If f :[a, b]→TFN (R) is a Lebesgue-integrable function, then f (t) dt is a fuzzy number. Proof By the Lebesgue-dominated convergence theorem (see Theorem 10.27 in [4]), we have b b b L R ˜ ˜ ˜ f (t) dt = f (t) dt, f (t) dt , h ∈ [0, 1]. h h a a a The result now follows by Theorem 1. Lemma 3 Let f :[a, b] → TFN(R) be a Lebesgue-integrable function. Then b b b b b L R L R f (t) dt = f (t) dt, f (t) dt, δ (t) dt, δ (t) dt , a a a a a where L R L R f (t) = ( f (t), f (t),δ (t),δ (t)). 12 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) Proof For any h ∈ [0, 1], we have b b b L R ˜ ˜ ˜ f (t) dt = f (t) dt, f (t) dt , h ∈ [0, 1], h h a a a and L L L L [ f (t)] = [ f (t)− (1− h)δ (t), f (t)+ (1− h)δ (t)], t ∈ [a, b]. Hence b b b L L R R f (t) dt = f (t)− (1− h)δ (t) dt, f (t)+ (1− h)δ (t) dt a a a b b b b L L R R = f (t) dt− (1− h) δ (t) dt, f (t) dt+ (1− h) δ (t) dt , a a a a which denotes a parameterized representation of a TFN with left center f (t) dt, b b b R L R right center f (t) dt, left spread δ (t) dt and right spread δ (t) dt. a a a Corollary 1 The value of the objective function of FSCLP, i.e., (˜γ+ ct ˜ ) x(t)dt, is a TFN for a given feasible solution (x(t), y(t)) whereγ ˜ and c ˜ are assumed to be TFNs. In the rest of the paper, we use the notation V (OP, x) to denote the objective func- tion value of an optimization problem (OP) with fuzzy valued objective function for a given feasible solution x with respect to a linear ranking function R. In particular, for a given feasible solution (x(t), y(t)) the objective function of FSCLP with respect to a linear ranking function R is denoted by V (FS CLP, (x(t), y(t))). Moreover, the nota- tion V (OP) is used to denote the optimum value of OP. It is assumed that V (OP) R R is ∞ if OP is an infeasible minimization problem and −∞ if OP is an infeasible maximization problem. We are now in a position to define an optimal solution for the FSCLP problem. n n 1) Definition 1 (Optimal Solution) Suppose that x(t) ∈ L [0, T ],y(t) ∈ C [0, T ] , and that R is a linear ranking function. The pair (x(t), y(t)) is said to be a feasible solution for FSCLP if it satisfies the set of constraints (1)-(3). Let Q (feasible region) be the set of all feasible solutions for FSCLP. We shall say that (x(t), y(t)) ∈ Qis optimal for FSCLP if V (FS CLP, (x(t), y(t)))≤ V (FS CLP, (¯ x(t), y ¯(t))), R R for all (¯ x(t), y ¯(t)) ∈ Q. 3. Discrete Approximations The algorithm that we describe in the next section solves successively two discrete approximations associated with FSCLP. In this section we present the mathematical formulation of these two discretizations and give some of their properties. 1) Notice that the notation L [0, T ] denotes the space of n dimensional vectors whose components are bounded measurable functions over [0, T ] and C [0, T ] denotes the space of n dimensional vectors whose components are continuous functions over [0, T ]. Fuzzy Inf. Eng. (2010) 1: 5-26 13 Given a partition P = {t ,..., t } of [0, T ] (i.e., 0 = t < t < ··· < t = T ), a 0 m 0 1 m fuzzy discrete approximation of FSCLP, denoted by FDP(P), is defined as follows: t + t k k−1 FDP(P) : min γ ˜ + c ˜ x ˆ(t +) k−1 k=1 s.t. (t − t )G x ˆ(t +)+ y ˆ(t ) = α+ at , 1 0 0 1 1 (t − t )G x ˆ(t +)+ y ˆ(t )− y ˆ(t ) = a(t − t ), k = 2,..., m, k k−1 k−1 k k−1 k k−1 H x ˆ(t +) ≤ b, k = 1,..., m, k−1 x ˆ(t ), y ˆ(t ) ≥ 0, k = 1,..., m. k−1 k The above problem is an instance of an ordinary fuzzy linear programming and efficiently can be solved by the method presented in [23,24]. Note that the labeling of the variables in FDP(P) is for convenience and does not mean that they explicitly refer to a function but rather in an implicit way as shown in the following lemma. Lemma 4 For any partition P, we have V (FS CLP) ≤ V (FDP(P)). R R Proof It is easy to see that any feasible solution for FDP(P) can be turned into a feasible solution for FSCLP with the same objective function value. Specifically, if (ˆ x, y ˆ) is a feasible solution for FDP(P), then ⎪ x ˆ(t +) k−1 , t ∈ [t , t ), k = 1,..., m, k−1 k t − t k k−1 x(t) = (4) x ˆ(t +) ⎪ m−1 , t = T, t − t m m−1 t − t t− t k k−1 y(t) = y ˆ(t )+ y ˆ(t ), t ∈ [t , t ], k = 1,..., m, (5) k−1 k k−1 k t − t t − t k k−1 k k−1 is the desired feasible solution for FSCLP. Lemma 4 shows that solving FDP(P) gives us an upper bound on the optimal value of FSCLP. To compute a lower bound on the optimal value of FSCLP, we consider another fuzzy discrete approximation of FSCLP which is a fuzzy variation of new discretization in Pullan [29]. For a given partition P = {t ,..., t }, the new discrete 0 m 14 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) approximation of FSCLP, denoted by FAP(P), is defined as: t + t k k−1 FAP(P) : min (γ ˜ + ct ˜ ) x ˆ(t +)+ (γ ˜ + ct ˜ ) x ˆ(t +) k−1 k−1 k k k=1 t − t t + t t + t 1 0 1 0 1 0 s.t. G x ˆ(t +)+ y ˆ = α+ a 2 2 2 t − t t + t k k−1 1 0 G x ˆ(t −)+ y ˆ(t )− y ˆ k k 2 2 t + t k k−1 = a t − , k = 1,..., m, t − t t + t k k−1 k k−1 G x ˆ(t +)+ y ˆ − y ˆ(t ) k−1 k−1 2 2 t + t k k−1 = a t , k = 2,..., m, k−1 H x ˆ(t +) ≤ b, k = 1,..., m, k−1 H x ˆ(t −) ≤ b, k = 1,..., m, t + t k k−1 x ˆ(t +), x ˆ(t −), y ˆ(t ), y ≥ 0, k = 1,..., m. k−1 k k In the sequel we present some properties of discretization FAP(P) that are needed for the purposes of this paper. Lemma 5 Let P be an arbitrary partition. Then FSCLP is feasible if and only if FAP(P) is feasible. Proof Let P = {t ,..., t } and (x ˆ, y ˆ) be a feasible solution for FAP(P). It is clear 0 m that this solution forms a feasible solution (x(t), y(t)) to FSCLP defined by 2ˆ x(t +) k−1 t −t ⎪ k k−1 , t ∈ t , , k = 1,..., m, k−1 ⎪ 2 t − t ⎪ k k−1 2ˆ x(t −) ⎨ k t −t k k−1 x(t) = , t ∈ , t , k = 1,..., m, (6) ⎪ k ⎪ 2 t − t k k−1 2ˆ x(t −) ⎪ m , t = T, t − t m m−1 with y(t) derived from the constraint (1). Now assume that (x(t), y(t)) is a feasible solution for FSCLP. Define (x ˆ, y ˆ)by t +t k−1 k x ˆ(t +) = x(t)dt, k = 1,..., m, (7) k−1 k−1 x ˆ(t −) = x(t)dt, k = 1,..., m, (8) t +t k−1 k y ˆ(t ) = y(t ), k = 1,..., m, (9) k k t + t t + t k k−1 k k−1 y ˆ = y , k = 1,..., m. (10) 2 2 Fuzzy Inf. Eng. (2010) 1: 5-26 15 Then, (x ˆ, y ˆ) is a feasible solution for FAP(P). Theorem 2 V (FAP(P)) ≤ V (FS CLP) holds for any partition P. R R Proof The proof is based on a particular dual problem FSCLP and hence the reader is referred to [25] for a detailed proof. Suppose that P = {t , t ,..., t } is a partition of the interval [0, T ] and (x ˆ, y ˆ)isan 0 1 m optimal solution for FAP(P). Let (x(t), y(t)) be the corresponding feasible solution for FSCLP constructed from (x ˆ, y ˆ) by (6). We define α[ˆ x, y ˆ]:=R (˜γ+ ct ˜ ) x(t) dt − ⎛ ⎞ (11) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R ⎜ (˜γ+ ct ˜ ) x ˆ(t +)+ (˜γ+ ct ˜ ) x ˆ(t −) ⎟ . k−1 k−1 k k ⎝ ⎠ k=1 The valueα[ˆ x, y ˆ] gives the difference in objective function values yielded by (x(t), y(t)) for FSCLP and (x ˆ, y ˆ) for FAP(P). By Theorem 2, we know that α[x, y] ≥ 0, and α[ˆ x, y ˆ] = 0 implies that (x(t), y(t)) is optimal for FSCLP. Theorem 3 Let {P } be any sequence of partitions such that lim ||P || = 0, n n→∞ n n=1 and (ˆ x , y ˆ ) be an optimal solution for F AP(P ). Then n n n lim α[ˆ x , y ˆ ] = 0. n n n→∞ Proof From Lemma 9 in [25], there is a constant K such that for any partition P, the following inequality holds: α[ˆ x , y ˆ ]≤||P||K, (12) P P where (x ˆ , y ˆ ) is an optimal solution for FAP(P) and||P|| is given by P P ||P|| := max{t − t : k = 1, 2...., m}. k k−1 The result now easily follows from the fact that α[ˆ x , y ˆ ] ≥ 0 for each partition P. P P 4. Constructing an Improved Solution In this section we show how to construct an improved solution for FSCLP, from a starting feasible (non-optimal) solution. This will form the basis of an algorithm for solving FSCLP. Let (x(t), y(t)) be a feasible solution for FSCLP in which x(t) is piecewise constant with breakpoints in a partition P = {t , t ,..., t }. Let (x ˆ, y ˆ) be the corresponding 0 1 m feasible solution for FAP(P) constructed from (x(t), y(t)) by (7)-(10). By Theorem 2, if (x ˆ, y ˆ) is an optimal solution for FAP(P), then (x(t), y(t)) is also an optimal solution ∗ ∗ for FSCLP. Otherwise, there exists a feasible solution, say (x ˆ , y ˆ ), for FAP(P) with strictly improved objective function value, i.e., ∗ ∗ δ[x(t), y(t)] = V (FAP(P), (ˆ x , y ˆ ))− V (FAP(P), (ˆ x, y ˆ)) < 0. (13) R R 16 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) ∗ ∗ It is not difficult to see that (x ˆ , y ˆ ) is also a feasible solution for FDP( P), where t + t t + t t + t 0 1 1 2 m−1 m P = t , , t , , t ,..., , t . 0 1 2 m 2 2 2 ∗ ∗ ∗ ∗ Hence, (x ˆ , y ˆ ) can be turned to a feasible solution (x (t), y (t)) for FSCLP by (6)-(7). The problem that we address here is how to construct a new feasible solution for FSCLP that has a better objective function value than the current feasible solution (x(t), y(t)). To achieve this aim, the first step is to construct a new partition P of [0, T ] by adding two breakpoints t + and t − in each interval [t , t ], where k−1 k k k k−1 k (t −t ) k−1 k = , and is a fixed value in [0, 1]. Thus, P is of the following form: P = {t , t + , t − , t + ,..., t − , t , t + ,..., t − , t }. 0 0 1 1 1 1 2 k k k k k+1 m m m The next step is to construct a solution (x ¯ (t), y ¯ (t)) for FSCLP with breakpoints in P by patching together the current feasible solution (x(t), y(t)) and the feasible ∗ ∗ ∗ ∗ solution (x ˆ , y ˆ ) constructed from (x ˆ , y ˆ ). Specifically, we define (x ¯ (t), y ¯ (t)) for each t ∈ [0, T]by ⎪ ∗ x (t), t ∈ [t , t + )∪ [t −, t ), k = 1,..., m, ⎨ k−1 k−1 i k i k x ¯ (t) = (14) x(t), otherwise, y ¯ (t) = a(t)− G x ¯ (s)ds. (15) Lemma 6 The solution (¯ x (t), y ¯ (t)), given by (14) and (15), is feasible for FSCLP. Proof The feasibility of (x ¯ (t), y ¯ (t)) follows from the fact that y ¯ is piecewise linear with breakpoints in partition P and y ¯ (t ) = (1−)y(t )+y ˜(t ), k = 1,..., m, k−1 k−1 k−1 t + t k k−1 y ¯ (t + ) = (1−)y(t )+y ˜ , k = 1,..., m, k−1 i k−1 t + t k k−1 y ¯ (t − ) = (1−)y(t )+y ˜ , k = 1,..., m. k i k See Lemma 4.1 in [29] for more details. The following theorem shows how to compute the change in the objective function value of FSCLP in moving from (x(t), y(t)) to (x ¯ (t), y ¯ (t)). Theorem 4 We have V (FS CLP, (¯ x (t), y ¯ (t)))− V (FS CLP, (x(t), y(t)))= (δ−α), R R ! " ∗ ∗ where α = α x ˆ , y ˆ andδ = δ[x(t), y(t)] are given by (11) and (13), respectively. Proof The result follows after some manipulation. As a direct consequence of this theorem, we observe that (x ¯ (t), y ¯ (t)) has a smaller objective function value than (x(t), y(t)) for appropriately chosen. Fuzzy Inf. Eng. (2010) 1: 5-26 17 Corollary 2 For small enough, (¯ x (t), y ¯ (t)) has a strictly smaller objective function value than x(t), y(t). Moreover, min V (FS CLP, (¯ x (t), y ¯ (t)))− V (FS CLP, (x(t), y(t))) R R ,α< 0 andδ< 2α, = 4α δ−α, otherwise, in which the minimum occurs at ⎪ ,α< 0 andδ< 2α, 2α = (16) 1, otherwise. Putting (x(t), y(t)) and (x ˜(t), y ˜(t)) together with = , given by (16), is referred to as patching together optimality. Theorem 5 Let (x(t), y(t)) be an optimal solution for FSCLP with x(t) piecewise con- stant with breakpoints in some P. Then the natural solution (ˆ x, y ˆ) is optimal for FAP(P). Proof Suppose that (x ˆ, y ˆ) is not optimal, then by patching together optimality, we can construct a feasible solution for FSCLP with strictly smaller objective function value (x(t), y(t)). This is a contradiction to the assumed optimality of (x(t), y(t)). This completes the proof. The results that we have obtained so far can be used to develop a new solution algorithm for FSCLP as given in Algorithm 1. It is worthwhile to mention that the basic idea of the algorithm is due to [29] for SCLP. Algorithm 1 (FSCLP Algorithm) Step 0 Set P = {0, T} as the initial partition. Construct an initial solution (x (t), y (t)) for FSCLP by constructing a feasible solution for FDP(P ). If 0 0 1 FDP(P ) is infeasible, then so is FSCLP (by Lemma 5). Set n = 1. Step 1 Let (x ˆ , y ˆ ) be the solution for FAP(P ) constructed from (x (t), y (t)). n−1 n−1 i n−1 n−1 If (x ˆ , y ˆ ) is an optimal solution for FAP(P ), then stop as (x (t), y (t)) is n−1 n−1 n n−1 n−1 optimal for FSCLP (by Theorem 2). ∗ ∗ Step 2 Solve FAP(P ) (by using the proposed method in [23,24]) to produce (x ˆ , y ˆ ). n n ∗ ∗ Let (x ˜ (t), y ˜ (t)) be the corresponding solution for FSCLP constructed from (x ˆ , y ˆ ). n n n n Step 3 Patch (x ˜ (t), y ˜ (t)) and (x (t), y (t)) together optimality to produce the so- n n n−1 n−1 lution (x ¯ (t), y ¯ (t)). Let P be the constructed partition from patching optimality n+1 n n together process. Step 4 Optimize FDP(P ) (by using the proposed method in [23,24]) to generate a n+1 solution (x (t), y (t)) to FSCLP. Set n = n+ 1 and return to Step 1. n n 18 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) We conclude this section with some remarks on the FSCLP Algorithm. The cur- rent form of the Algorithm 1 may be not practical because at each iteration the number of breakpoints is increased by a factor of 2 or 3. In the rest of this section, we briefly discuss how to address this problem. Removing Redundant Breakpoints. Shortly after the development of Pullan’s al- gorithm in [29] for SCLP, Philpott and Craddock [28] used the ideas of [29] to pro- duce an algorithm for solving continuous-time network flows (but with a direct exten- sion to include SCLP). It is a discretization algorithm which proceeds by adding and removing points in the partition used in the current discretization. This method can be extended for FSCLP without any difficulties. In fact, it is possible that some break- points can be removed from the partition and some adjacent intervals can be merged while improving the solution value. The reduction of unnecessary breakpoints in the solution is a key feature since they increase the size of the subproblems to be solved at each iteration, leading to long computation times. To clarify the discussion, let (ˆ x , y ˆ ) be an optimal solution for FDP(P ) generated at iteration n of FSCLP Algo- n n n rithm and (x (t), y (t)) denotes the associated solution for FSCLP. Usually, the x (t)is n n n identical in some consecutive intervals of P and as a consequence, some breakpoints are redundant. Specifically, a breakpoint t in P is said to be redundant if k n x ˆ (t +) x ˆ (t +) n k−1 n k = . t − t t − t k k−1 k+1 k It is clear that if t is redundant, then it can be removed from P without increasing k n the objective function value. Thus, it is reasonable to remove the redundant break- points because they not only increase the size of the subproblems, leading to long computation times, but also obscure the structure of optimal solutions. 5. Numerical Examples As mentioned previously, SCLP arises in a number of application areas such as traf- fic control, production systems, communication networks, and pipeline systems for transporting (see, for instance, [17]). In particular, it can be used to model dynamic network flow problems (single or multicommodity) where the variables are rates of flow, the costs and demands are time varying and storage is permitted at the nodes of the network. In this section we consider three dynamic network flow examples taken from Philpott and Craddock [28]. But in contrast to [28], we assume that arc costs are subjected to uncertainty and are given by fuzzy valued functions. These three exam- ples are solved by using the uniform discretization algorithm presented in [25], FS- CLP Algorithm, and FSCLP Algorithm with removing redundant breakpoints. For each algorithm, we use the Yager’s method [37] as the linear ranking function R for comparison of fuzzy numbers and give the results of the first five iterations. In partic- ular, at each iteration we report optimum values of V (FDP(P )) and V (FAP(P )), R n R n the error bound δ := V (FAP(P )) − V (FDP(P )), the number of breakpoints in n R n R n P , denoted by “# BP”, and the number of the breakpoints after removing redundant breakpoints at optimum solution of FDP(P ), denoted by “# BPR”. n Fuzzy Inf. Eng. (2010) 1: 5-26 19 1 4 1 2 1 3 4 1 3 4 3 4 2 5 Fig. 1 Network for Example 1 Fig. 2 Network for Example 2 Example 1 In the first example, we consider a network with four nodes (numbered from 1 to 4) and four arcs (numbered from 1 to 4) connecting those nodes as shown in Fig. 1. Each arc j has a capacity b (t), given by: b (t) = 0.6, b (t) = 0.8, 1 2 b (t) = 0.8, b (t) = 1.6. 3 4 In fact, b (t) is an upper bound of the rate of flow in arc j at time t. Moreover, each arc has a cost per unit flow, which is time-dependent. The problem here is to send an initial storage of 8 units from node 1 into node 4 over the interval [0, 10] so that the cost is minimized. This problem can be formulated as an instance of SCLP problem by putting a constant demand of 1.6 per unit time at node 4 during (5, 10]. In term of SCLP problem, the decision variable x represents the rate of flow entering arc j at time t and y (t) represents the amount of flow stored at node i at time t. Moreover, G is the node-arc incidence matrix of the network and H is an identity matrix. More specifically: T = 10, n = n = n = 4, 1 2 3 ⎡ ⎤ ⎡ ⎤ ⎢ 100 0⎥ ⎢ 10 00⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −11 0 0 01 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ G = ⎢ ⎥ , H = ⎢ ⎥ , ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −1 −11 00 1 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 00 0 −1 00 0 1 (t)isgiven by and a 0, t ∈ [0, 5], a (t) = 8, a (t) = a (t) = 0, a (t) = 1 2 3 4 −1.6(t− 5), t ∈ [5, 10]. For each node i, the value a (t) can be interpreted as the amount of available supply or required demand at node i up to time t, depending on whether a (t) is positive or negative. We assume that transit costs are subjected to uncertainty and they are expressed by 20 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) TFNs as follows: c ˜ (t) = (0.5, 1.5, 0.6, 0.8)+ (0.4, 0.8, 0.2, 0.6)t, c ˜ (t) = (0.8, 1.2, 0.5, 0.5)+ (0.8, 2.2, 1, 1.2)t, c ˜ (t) = (10, 14, 2, 1)− t, c ˜ (t) = (5.5, 6.5, 0.8, 0.5)− 0.2t, L R L R where (a , a ,α ,α ) denotes a TFN. The computational sequence is shown in Table 1. After five iterations, the uniform discretization algorithm, FSCLP Algorithm, and FSCLP Algorithm with removing redundant breakpoints give us approximation solutions with breakpoints in {0, 5, 5.3125, 5.9375, 6.25, 10}, {0, 4.9726, 4.9734, 4.9821, 4.9829, 5, 5.0529, 5.3293, 6.0189, 10}, {0, 5, 5.0193, 5.3554, 6.0110, 10}, such that the error bounds are guaranteed to be less than 0.0485, 0.0075, and 0.0200 respectively. We notice that although the solution obtained by the FSCLP Algorithm is more accurate than the two other algorithms, it has more breakpoints. Table 1: Test results for Example 1. n V (FDP(P )) V (FAP(P )) δ # BP # BPR R n R n n Uniform Discretization Algorithm 1 86.6000 79.7500 6.8500 2 2 2 85.4875 82.3000 3.1875 4 3 3 83.8938 83.6547 0.2391 8 3 4 83.8937 83.7617 0.1320 16 3 5 83.8694 83.8209 0.0485 32 5 FSCLP Algorithm 1 86.6000 79.7500 6.8500 2 2 2 83.9197 83.7339 0.1858 6 3 3 83.8851 83.8327 0.0524 18 7 4 83.8531 83.8358 0.0173 54 9 5 83.8514 83.8439 0.0075 162 9 FSCLP Algorithm with removing redundant breakpoints 1 86.6000 79.7500 6.8500 2 2 2 83.9197 83.5269 0.3928 3 3 3 83.8926 83.8157 0.0769 5 5 4 83.8884 83.8320 0.0564 7 7 5 83.8538 83.8338 0.0200 5 5 Fuzzy Inf. Eng. (2010) 1: 5-26 21 Example 2 The second example is depicted in the network shown in Fig. 2. The functions describing the flow bounds and supplies/demands are as follows: b (t) = b (t) = b (t) = b (t) = 2, b (t) = 1, 1 2 4 5 3 ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ 4t, t ∈ [0, 5], t, t ∈ [0, 5], ⎨ ⎨ a (t) = a (t) = ⎪ ⎪ 1 2 ⎪ ⎪ ⎩ ⎩ 2t+ 10, t ∈ (5, 10], 2t− 5, t ∈ (5, 10], −t, t ∈ [0, 5], a (t) = a (t) = −3t. 3 4 5− 2t, t ∈ (5, 10], The arc costs are given by the following fuzzy valued functions: c ˜ (t) = (8, 12, 0.5, 0.7)− (0.4, 0.8, 0.2, 0.2)t, c ˜ (t) = (6, 8, 0.4, 0.4), c ˜ (t) = (5, 7, 0.3, 0.5)− 6t, c ˜ (t) = (1, 3, 0.4, 0.2)+ (0.5, 1.5, 0.2, 0.2)t, c ˜ (t) = (2, 6, 1, 1). The computational result is reported in Table 2. The uniform discretization algo- rithm generates an approximate algorithm with breakpoints in {0, 3.75, 4.6875, 5, 5.3125, 5.625, 5.9375, 6.25, 6.875, 7.1875, 7.5, 7.8125, 8.4375, 8.75, 9.0625, 10}, such that the error bounds are guaranteed to be less than 0.0156. For the two other ∗ ∗ algorithms an optimal solution (x , y ) is obtained at the second iteration with break- points in the partition{0, 3.6875, 5.0000, 8.6875, 10}. Table 2: Test results for Example 2. n V (FDP(P )) V (FAP(P )) δ # BP # BPR R n R n n Uniform Discretization Algorithm 1 397.7500 392.5000 5.2500 2 2 2 397.5000 395.1250 2.3750 4 4 3 396.3750 396.3125 0.0625 8 4 4 396.3750 396.3438 0.0312 16 9 5 396.3750 396.3594 0.0156 32 15 FSCLP Algorithm 1 397.7500 392.5000 5.2500 2 2 2 397.3719 396.3719 0.0000 6 5 FSCLP Algorithm with removing redundant breakpoints 1 397.7500 392.5000 5.2500 2 2 2 397.3719 396.3719 0.0000 5 5 22 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) Example 3 The second example is posed in the network shown in Fig. 3. The flow bounds and demands are as follows: b (t) = b (t) = 1, b (t) = b (t) = 2, 1 2 3 4 b (t) = b (t) = 1, b (t) = b (t) = 2, 5 6 7 8 b (t) = b (t) = 1, b (t) = b (t) = 3, 9 10 11 12 b (t) = b (t) = 4, 13 14 ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ 0.4t, t ∈ [0, 5], 1.6t, t ∈ [0, 5], ⎨ ⎨ a (t) = a (t) = 1 ⎪ 2 ⎪ ⎪ ⎪ ⎩ ⎩ 2, t ∈ (5, 10], 8, t ∈ (5, 10], ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ 0.4t, t ∈ [0, 5], 0.4t, t ∈ [0, 5], ⎨ ⎨ a (t) = a (t) = 3 ⎪ 4 ⎪ ⎪ ⎪ ⎩ ⎩ 2, t ∈ (5, 10], 2, t ∈ (5, 10], ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ 1.6t, t ∈ [0, 5], 0.4t, t ∈ [0, 5], ⎨ ⎨ a (t) = a (t) = 5 ⎪ 6 ⎪ ⎪ ⎪ ⎩ ⎩ 8, t ∈ (5, 10], 2, t ∈ (5, 10], ⎪ 2t, t ∈ [0, 5], a (t) = 0, t ∈ [0, 10], a (t) = 7 8 ⎪ 10, t ∈ (5, 10], a (t) = 0, t ∈ [0, 10], a (t) = 0, t ∈ [0, 10], 9 10 0, t ∈ [0, 5], a (t) = a (t) = 0, t ∈ [0, 10]. 11 ⎪ 12 −6.8t, t ∈ (5, 10], The arc costs are given by the following fuzzy valued functions: c ˜ (t) = (6, 8, 0.5, 0.8), c ˜ (t) = (0.5, 1.5, 0.2, 0.4)+ (1, 1.8, 0.3, 0.3)t, c ˜ (t) = (4, 6, 0.5, 0.5)+ (0.4, 0.8, 0.3, 0.3)t, c ˜ (t) = (9, 11, 0.6, 0.8)− (0.3, 1.3, 0.2, 0.4)t, c ˜ (t) = (10, 14, 1, 1), c ˜ (t) = (7, 11, 1.5, 2), c ˜ (t) = (10, 14, 1.5, 1.2)− (0.4, 0.8, 0.2, 0.2)t, c ˜ (t) = 5+ (0.5, 0.7, 0.2, 0.2)t, c ˜ (t) = (8, 10, 1, 1), c ˜ (t) = (5, 7, 0.8, 1), c ˜ (t) = (0.5, 1.5, 0.3, 0.3)t. c ˜ (t) = (2, 6, 0.5, 0.7), c ˜ (t) = (0.8, 1.2, 0.2, 0.2), c ˜ (t) = (0.6, 1.4, 0.2, 0.2). 14 Fuzzy Inf. Eng. (2010) 1: 5-26 23 1 2 1 2 3 3 4 5 6 4 5 6 7 8 9 10 7 8 9 11 12 13 14 10 11 12 Fig. 3 Network for Example 3 The computational sequence is given in Table 3. In this example, the number of breakpoints at the solutions generated by all three algorithms grows exponentially in the number of iterations. Table 3: Computational results for Example 3. n V (FDP(P )) V (FAP(P )) δ # BP # BPR R n R n n Uniform Discretization Algorithm 1 580.0250 574.5250 5.500 2 2 2 579.0875 577.1187 1.9688 4 3 3 578.6219 577.8188 0.8031 8 8 4 578.3500 578.1777 0.1723 16 15 5 578.2973 578.2346 0.0627 32 30 FSCLP Algorithm 1 580.0250 574.5250 5.500 2 2 2 579.5215 576.0888 3.4327 6 6 3 578.8214 577.7672 1.0542 18 18 4 587.4330 587.1814 0.2516 54 52 5 578.3128 578.2426 0.0702 162 142 FSCLP Algorithm with removing redundant breakpoints 1 580.0250 574.5250 5.500 2 2 2 579.5215 576.0888 3.4327 6 6 3 578.8214 577.7672 1.0542 18 18 4 587.4330 587.1814 0.2516 52 52 5 578.2775 578.2626 0.0150 131 131 24 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) The growth of the number of breakpoints is used to compare the performance of the algorithms. As computational results show, FSCLP Algorithm with removing re- dundant breakpoints yields reasonable solutions and is clearly superior to the uniform discretization algorithm and the FSCLP Algorithm since the number of breakpoints grows considerably faster for the latter algorithms. 6. Conclusion In this paper we studied separated continuous linear programs with fuzzy valued ob- jective functions, so-called FSCLP. We followed the ideas in Pullan [30] to construct an improved solution for FSCLP, from a starting feasible (non-optimal) solution for FSCLP. This improvement step allowed us to develop a solution algorithm for FSCLP, the so-called FSCLP Algorithm. To enhance the performance of this algorithm, we used the idea of removing redundant breakpoints at each iteration as they increase the size of subproblems to be solved. We applied the uniform discretization algo- rithm developed in [25], FSCLP Algorithm, and FSCLP Algorithm with removing redundant breakpoints to solve three continuous-time network flow problems with fuzzy valued objective functions. The computational results showed that the FSCLP Algorithm with removing redundant breakpoints generates better solutions with less breakpoints than those produced by the uniform discretization algorithm [25] and FSCLP Algorithm. We end the paper by mentioning that although, to simplify notation, we assumed that the problem data (i.e. the vectorsγ, ˜ c ˜,α, a, b) are constant, all the results can be readily generalized to the case that they are piecewise constants over [0, T ]. References 1. Anderson E J (1978) A continuous model for job-shop scheduling. U.K. : PhD. Thesis, University of Cambridge 2. Anderson E J, Nash P, Perold A F (1983) Some properties of a class of continuous linear programs. SIAM J. Control and Optimization 21: 258–265 3. Anderson E J, Philpott A B (1994) On the solutions of a class of continuous linear programs. SIAM J. Control and Optimization 32: 1289-1296 4. Apostol T M (1974) Mathematical analysis (2th edition). California: Addison-Wesley 5. Bellman R E (1953) Bottleneck problem and dynamic programming. Proc. Nat. Acad. Sci. 39: 947–951 6. Bellman R E (1957) Dynamic programming. Princeton University Press, NJ 7. Bellman R E, Zadeh L A (1970) Decision-making in a fuzzy environment. Management Science 17: 141–164 8. Buckley J J (1988) Possibilistic linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems 26: 135-138 9. Buckley J J (1989) Solving possibilistic linear programming. Fuzzy Sets and Systems 31: 329-341 10. Buie R N, Abraham J (1973) Numerical solutions to continuous linear programming problems. Zeitschrif fur ¨ Operations Research 17: 107-117 11. Chen L H, Ko W C (2009) Fuzzy linear programming models for new product design using QFD with FMEA. Applied Mathematical Modelling 33: 633-647 12. Ernst A T (1996) Continuous time quadratic cost flow problems with applications to water distribution networks. Journal of the Australian Applied Mathematical Society 37: 530-548 13. Fleischer L, Sethuraman J (2005) Efficient algorithms for separated continuous linear programs: The multi-commodity flow problem with holding costs and extensions. Mathematics of Operations Re- search 30: 916-938 Fuzzy Inf. Eng. (2010) 1: 5-26 25 14. Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets and Systems 18: 31-43 15. Hajek B, Ogier R G (1984) Optimal dynamic routing in communication networks with continuous traffic. Networks 14: 457-487 16. Jimenez M, Arenas M, Bilbao A, Rodriguez M V (2007) Linear programming with fuzzy parameters, An interactive method resolution. European Journal of Operational Research 177: 1599-1609 17. Luo X D (1995) Continuous linear programming: Theory, algorithms and applications. PhD Thesis, Cambridge MA: MIT Operations Research Center 18. B. Julien (1994) A extension to possibilistic linear programming. Fuzzy Sets and Systems 64: 195- 19. Kim Y K, Ghil B M (1997) Integrals of fuzzy-number-valued functions. Fuzzy Sets and Systems 86: 213-222 20. Lai Y J, Hwang C L (1992) A new approach to some possibilistic linear programming problem. Fuzzy Sets and Systems 49: 121-133 21. Liu X (2001) Measuring the satisfaction of constraints in fuzzy linear programming. Fuzzy Sets and Systems 122: 263-275 22. Maleki H R (2002) Ranking functions and their applications to fuzzy linear programming. Far East Journal of Mathematical Sciences 4: 283-301 23. Maleki H R, Tata M, Mashinchi M (2000) Linear programming with fuzzy variables. Fuzzy Sets and Systems 109: 21-33 24. Mahdavi-Amiri N, Nasseri S H (2007) Duality results and a dual simplex method for linear program- ming problems with trapezoidal fuzzy variables. Fuzzy Sets and Systems 158: 1961-1978 25. Nasrabadi M M, Yaghoobi M A, Mashinchi M (2009) An approximation algorithm for separated th continuous linear programs with fuzzy valued objective functions. Proceedings of the 40 Annual Iranian Mathematics Conference, 17-20 August 2009, Tehran, Iran, to appear (A PDF file can be found at http://aimc40.ir/sites/aimc40.ir/files/papers/pdf/399.pdf) 26. Nazarathy Y, Weiss G (2009) Near optimal control of queueing networks over a finite time horizon. Annals of Operations Research, 170: 233-249 27. Peidro D, Mula J, Jimenez M, Botella M M (2009) A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment. European Journal of Operational Research, In Press 28. Philpott A B, Craddock M (1995) An adaptive discretization algorithm for a class of continuous network programs. Networks 261-11 29. Pullan M C (1993) An algorithm for a class of continuous linear programs. SIAM J. Control and Optimization 31: 1558-1577 30. Pullan M C (1995) Forms of optimal solutions for separated continuous linear programs. SIAM J. Control and Optimization 33: 1952-1977 31. Pullan M C (1996) A duality theory for separated continuous linear programs. SIAM J. Control and Optimization 34: 931-965 32. Pullan M C (1997) Existence and duality theory for separated cnotinuous linear programs. Math. Modelling Systems 3: 219-245 33. Rommelfanger H (1996) Fuzzy linear programming and its applications. European Journal of Oper- ational Research 92: 512-527 34. Roubens M (1991) Inequality constraints between fuzzy numbers and their use in mathematical pro- gramming, in: R. Slowinsky and J. Teghem, Eds.. Stochastic Versus Fuzzy Approaches to Multiob- jective Mathematical Programming Under Uncertainty, Kluwer Academic Publishers: 321-330 35. Shaocheng T (1994) Interval number and fuzzy number linear programming. Fuzzy Sets and Systems 66: 301-306 36. Tanaka H, Guo P, Zimmermann H J (2000) Possibility distributions of fuzzy decision variables ob- tained from possibilistic linear programming problems. Fuzzy Sets and Systems 113: 323-332 37. Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval. Information Sciences 24: 143-161 38. Vasant P M (2003) Application of fuzzy linear programming in production planning. Fuzzy Opti- mization and Decision Making 2: 229-241 26 Mohammad Mehdi Nasrabadi · Mohammad Ali Yaghoobi · Mashaallah Mashinchi (2010) 39. Wang X, Kerre E E (2001) Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems 118: 375-385 40. Wang R C, Liang T F (2003) Application of fuzzy multi-objective linear programming to aggregate production planning. Computers and Industrial Engineering 46: 17-41 41. Wu C, Gong Z (2000) On Henstock integrals of interval-valued functions and fuzzy-valued functions. Fuzzy Sets and Systems 115: 377-391 42. Wu C, Gong Z (2001) On Henstock integral of fuzzy-number-valued functions (I). Fuzzy Sets and Systems 120: 523-532 43. Zimmermann H J (1985) Fuzzy sets theory and its applications. Boston: Kluwer Academic Publishers
Fuzzy Information and Engineering – Taylor & Francis
Published: Mar 1, 2010
Keywords: Continuous linear programming; Fuzzy linear programming; Duality; Discretization
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