Using gradual numbers for solving fuzzy-valued combinatorial optimization problems⋆ Adam Kasperski1 and Pawel Zieli´ nski2 1
Institute of Industrial Engineering and Management, Wroclaw University of Technology, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland,
[email protected] 2 Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland,
[email protected]
Abstract. In this paper a general approach to combinatorial optimization problems with fuzzy weights is discussed. The results, valid for the interval-valued problems, are extended to the fuzzy-valued ones by exploiting the very recent notion of a gradual number. Some methods for determining the exact degrees of possible and necessary optimality and the possibility distributions of deviations of solutions and elements are proposed. The introduced notions are illustrated by practical examples.
1
Introduction
In a combinatorial optimization problem we are given a finite set of elements E and a weight we is associated with each element e ∈ E. We seek an object composed of the elements of E whose total weight is minimal or maximal. In this case the solutions and elements can be divided into two groups: optimal and non-optimal ones. One may also evaluate the optimality from the point of view of a deviation, that describes how far a solution (an element) is from being optimal. In this paper, we consider the case in which the element weights are ill-known and they are modeled by means of intervals or fuzzy intervals. In the interval-valued case solutions and elements form three groups: those that are optimal for sure (necessarily optimal ), those that are not optimal for sure and those, whose optimality is unknown (possibly optimal ). Now, to each solution and element an interval of possible deviations from optimum may be associated. In the fuzzy-valued case the notions of optimality can be extended and every solution (element) can be characterized by degrees of possible and necessary optimality and a possibility distribution of its deviation. There exists a wide class of combinatorial optimization problems for which the characterization of optimality of solutions and elements can be efficiently done in the interval model. In this paper we extend these results to the fuzzyvalued case by exploiting a very recent notion of a gradual number, which provides a new outlook on fuzzy intervals. This allows to apply interval methods ⋆
This work was partially supported by Polish Committee for Scientific Research, grant 3T11C05430.
to problems with fuzzy weights. In consequence, we propose some methods for determining the exact degrees of possible and necessary optimality and the possibility distributions of deviations of solutions and elements for some fuzzy-valued combinatorial optimization problems.
2
Preliminaries
Let E = {e1 , . . . , en } be a finite ground set and let Φ ⊆ 2E be a set of subsets of E called the set of the feasible solutions. In a deterministic case, for every element e ∈ E there is given a nonnegative real weight we . A combinatorial optimization problem P with a linear objective function consists in finding a feasible solution X ∈ Φ whose total weight is minimal, namely: X we . (1) P : F ∗ = min F (X) = min X∈Φ
X∈Φ
e∈X
Formulation (1) encompasses a large variety of deterministic combinatorial optimization problems, for instance shortest path, minimum spanning tree, minimum assignment, 0-1 knapsack etc. Some of them are polynomially solvable while the other ones are NP-hard. For a wide review we refer the reader to [11]. A solution that minimizes (1) is said to be optimal. We also call an element e ∈ E optimal if it is a part of an optimal solution. Thus, in the deterministic case, the set of solutions Φ and the set of elements E can be divided into two groups: the optimal and non-optimal ones. In the case when solution X ∈ Φ or element f ∈ E are not optimal, a natural question arises: how far they are from optimality. One can define a deviation δX of solution X in the following way: X X δX = F (X) − F ∗ = we . we − min e∈X
X∈Φ
e∈X
Similarly, a deviation δf of element f is defined as follows: X X we , δf = Ff∗ − F ∗ = min we − min X∈Φf
e∈X
X∈Φ
e∈X
where Φf ⊆ Φ is the set of all solutions containing f . In other words, deviation δX (resp. δf ) describes how far solution X (resp. element f ) is from being optimal. Obviously, solution X (element f ) is optimal if and only if δX = 0 (δf = 0).
3
Interval-valued combinatorial optimization problem
Suppose that the values of the element weights are not precisely known, but they are known to belong to intervals We = [we− , we+ ], e ∈ E. It means that the actual value of a weight will take some value within an interval, but it is not possible to predict at present which one. A configuration is a precise instantiation of the weights of each element w = (we )e∈E , we ∈ We . Thus, every configuration
expresses a realization of the weights which may occur. We denote by Γ the set w ) to denote the weight of all configurations, i.e. Γ = ×e∈E [we− , we+ ]. We use we (w of element e ∈ E in a specified configuration w ∈ Γ . Among the configurations of Γ we distinguish the extreme ones, which belong to ×e∈E {we− , we+ }. Let A be a given subset of the element set E. In extreme + configuration w + A all elements e ∈ A have weights we and all the remaining − elements have weights we . Similarly, in extreme configuration w − A all elements e ∈ A have weights we− and all the remaining elements have weights we+ . These extreme configurations will play a crucial role in further considerations. For a given solutionPX ∈ Φ, we define its weight under a fixed configuration w ). We will denote by F ∗ (w w ) the value of the w ∈ Γ as F (X, w ) = e∈X we (w weight of an optimal solution under w ∈ Γ , that is w ) = min F (X, w ). F ∗ (w X∈Φ
(2)
Notice that if configuration w is fixed, then (2) is problem P. Hence the optimality of a given solution or element now depends on configuration w and the following characterization can be provided: a solution X ∈ Φ (element f ∈ E) is said to be possibly optimal if there exists a weight configuration w ∈ Γ for which it is optimal; a solution X (element f ) is said to be necessarily optimal if it is optimal for all weights configurations w ∈ Γ . As in the deterministic case, we can obtain an additional information about the optimality of a given solution or element using the concept of a deviation. w ) and δf (w w ) a deviation of solution X, element f , in configuration Denote by δX (w − + w . We can now define the widest interval ∆X = [δX , δX ] of possible values of deviations of solution X in the following way: − + w ) and δX w ). δX = min δX (w = max δX (w w ∈Γ
w ∈Γ
(3)
In the same way, we define the widest interval ∆f = [δf− , δf+ ] of possible values of deviation of element f . Intervals ∆X and ∆f measure how far X and f are from being possibly (resp. necessarily) optimal. It is clear that solution X is − possibly optimal if and only if δX = 0 and it is necessarily optimal if and only if + δX = 0. The same relations hold for elements. The following proposition shows the crucial role of the extreme configurations: w ) and δf (w w ) over all conProposition 1. The maximum and minimum of δX (w figurations w ∈ Γ are attained in extreme configurations from set ×e∈E {we− , we+ }. Moreover, the following equalities hold for every solution X: − − ∗ w− w− = δX (w δX X ), X ) = F (X, w X ) − F (w + + + ∗ w X ) = F (X, w X ) − F (w w+ δX = δX (w X ).
(4) (5)
Proof. The fact that the minimum and maximum are attained in extreme configurations follows from the results obtained in [10] and the property that funcw ) and δf (w w ) are locally monotonic with respect to each variable tions δX (w
we ∈ [we− , we+ ], e ∈ E. In order to prove (4) suppose that w is a configuraw ) and denote by X ∗ the optimal solution in configtion that minimizes δX (w − − w ) = F (X, w ) − F ∗ (w w ) ≥ F (X, w ) − F (X ∗ , w ) ≥ uration w X . It holds δX (w − − − ∗ ∗ ∗ w− F (X, w X ) − F (X , w X ) = F (X, w X ) − F (w X ). Hence, configuration w − X also w ). The proof of equality (5) is similar. minimizes δX (w ⊓ ⊔ From Proposition 1 it follows that if problem P is polynomially solvable, then interval ∆X for a given solution X can be computed in polynomial time. Hence, the possible and necessary optimality of X can be characterized in polynomial time as well. Contrary to the optimality of solutions, there is no an easy characterization of the optimality of the elements. We know that configuration w ) is an extreme one, but it may be hard to w that minimizes or maximizes δf (w compute (observe that there are up to 2n extreme configurations). The complexity of computing interval ∆f , for a given element f ∈ E, strongly depends on a particular problem P and it may be NP-hard even if P is polynomially solvable. For instance, for Shortest Path, determining a configuration that minimizes w ) is strongly NP-hard [1], but determining a configuration that maximizes δf (w w ) is polynomially solvable in acyclic digraphs [7]. δf (w
4
Fuzzy-valued combinatorial optimization problem
In this section we extend the notion of a deviation of a solution and element to the more general fuzzy case. In consequence, we also generalize the notions of possible and necessary optimality of solutions and elements. In order to compute the fuzzy deviations we apply a recent concept of a gradual number [8]. 4.1
Gradual numbers and fuzzy intervals
The classical intervals model uncertainty in a Boolean way: a value in the interval is possible; a value outside is impossible. The idea of fuzziness is to move from the Boolean way to a gradual one. Hence fuzziness makes the boundaries of the interval softer and thus making uncertainty gradual. In order to model the essence of graduality without uncertainty the concept of a gradual number has been recently proposed. Following the notation of [8] a gradual real number (or gradual number for short) r˜ is defined by an assignment function Ar˜ from (0, 1] to IR. It can be seen as a number parametrized by a value of λ ∈ (0, 1]. Making use of the notion of a gradual number, one can describe a fuzzy in˜ by an ordered pair of two gradual numbers [w terval W ˜− , w ˜ + ], where w ˜ − is a + ˜ and w ˜ . In order to gradual lower bound of W ˜ is a gradual upper bound of W ensure the well known shape of a fuzzy interval, w ˜ − and w ˜ + must satisfy the following properties: Aw˜ − is an increasing function; Aw˜ + is a decreasing function and Aw˜ − (1) ≤ Aw˜ + (1). In this paper we will additionally assume that the assignment functions of gradual numbers are continuous and their domains are extended to interval [0, 1]. In consequence, the corresponding membership functions of fuzzy intervals are continuous and have a compact support. Observe that
a fuzzy interval can be now viewed as an interval of gradual numbers bounded by w ˜ − and w ˜ + (see Fig. 1). For a deeper discussion on gradual numbers and their relationships with fuzzy intervals, we refer the reader to [4, 5, 8]. µW ˜ (x) 1
w ˜−
w ˜+
x ˜ (in bold). Fig. 1. The left and right bounds of fuzzy interval W
The classical arithmetic operations on gradual numbers are defined by operations on their assignment functions. Let r˜ and s˜ be two gradual numbers. The sum of r˜ and s˜ is defined by summing their assignment functions, that is Ar˜+˜s (λ) = Ar˜(λ) + As˜(λ) for all λ ∈ (0, 1]. The subtraction, maximum and minimum of gradual numbers can be defined in a similar manner. Observe that the minimum and maximum operations on gradual numbers are not selective, that is in general case max(Ar˜, As˜) 6= Ar˜ or As˜. There are subintervals of (0, 1] where max(Ar˜, As˜) = Ar˜ and it is As˜ in the complementary subinterval. It is worth pointing out that most algebraic properties of real numbers are preserved for gradual real numbers, contrary to the case of fuzzy intervals. A concept of a fuzzy interval of the L-R type (see, e.g. [2]) is very popular and convenient in applications. A fuzzy interval of the L-R type is a fuzzy set in the space of real numbers, whose membership function is of the following form: 1 for x ∈ [w− , w+ ], w− −x L αW for x ≤ w− , µW ˜ (x) = + R x−w for x ≥ w+ , βW
where L and R are continuous non-increasing functions, defined on [0, +∞), strictly decreasing to zero in those subintervals of the interval [0, +∞) in which they are positive, and fulfilling the conditions L(0) = R(0) = 1. The parameters αW and βW are non-negative real numbers. A special case of a fuzzy interval is a triangular fuzzy interval in which L(x) = R(x) = max{0, 1 − x} and w− = w+ . ˜ of the L-R type can be It is denoted by triple (w, αW , βW ). A fuzzy interval W − described by an ordered pair of gradual numbers [w ˜ ,w ˜ + ] with the following assignment functions: Aw˜ − (λ) = w− − L−1 (λ)αW and Aw˜ + (λ) = w+ + R−1 (λ)βW ,
(6)
where L−1 (similarly R−1 ) denotes the inverse function to L in this part of its domain in which it is positive.
4.2
A possibilistic formulation of the problem
Assume that the element weights are ill-known, uncontrollable and unrelated parameters we , e ∈ E, with fuzzy sets of more or less possible values. We say ˜ e ”, where W ˜ e is a fuzzy interval associated that the assertion of the form “we is W with we , generates the possibility distribution of we with respect to the formula Π(we = x) = µW ˜ e (x). For the interpretation of the possibility distribution we refer the reader to [2]. A configuration w = (we )e∈E represents a state of the world, where we = we for every e ∈ E. The joint possibility distribution over all configurations is as follows: w ) = Π(∧e∈E (we = we )) = min Π(we = we ) = min µW π(w ˜ e (we ). e∈E
e∈E
We can now fuzzyfy intervals ∆X and ∆f obtaining fuzzy intervals ∆˜X and ∆˜f , whose membership functions express the possibility distributions for deviations δX and δf . This can be done in the following way: µ∆˜X (δ) = Π(δX = δ) = µ∆˜f (δ) = Π(δf = δ) =
w ), π(w
sup w : δX (w w )=δ} {w
sup
w ). π(w
w : δf (w w )=δ} {w
Having the possibility distribution for ∆˜X we can calculate the possibility and necessity of an event δX ∈ [δ − , δ + ] in the following way: Π(δX ∈ [δ − , δ + ]) =
sup δ∈[δ − ,δ + ]
µ∆˜X (δ),
N(δX ∈ [δ − , δ + ]) = 1 − Π(δX ∈ / [δ − , δ + ]) = 1 −
sup δ ∈[δ / − ,δ + ]
µ∆˜X (δ).
The same formulae hold for an element deviation δf . Considering the particular event δX = 0 we can compute the degrees of possible and necessary optimality of solution X in the following way: Π(X is optimal) = Π(δX = 0) = µ∆˜X (0),
(7)
N(X is optimal) = N(δX = 0) = 1 − Π(δX > 0) = 1 − sup µ∆˜X (δ).
(8)
δ>0
In the same way we define the degrees of possible and necessary optimality of element f , using ∆˜f . 4.3
˜X and ∆ ˜f Application of gradual numbers to computing ∆
In this section we show that the concept of a gradual number allows to extend naturally all the results from Section 3 to the fuzzy-valued case. In consequence we obtain methods for computing the fuzzy deviations ∆˜X and ∆˜f . We start with introducing the notion of a fuzzy configuration, which is a vector (w w) ˜ e∈E of gradual numbers specified for the weight of every element e ∈ E. Now, the
˜ is F˜ (X, w ˜) = gradual weight of a solution X ∈ Φ under a fuzzy configuration w P ∗ ˜ ˜ ˜ ˜ ˜ ∗ (w ˜ ˜ ˜ ˜ ˜ ˜) w ˜ ( w ) and F ( w ) = min F (X, w ). Hence δ ( w ) = F (X, w ) − F e X∈Φ X e∈X P ˜ ) − F˜ ∗ (w ˜ ). Observe, that ˜ ) = F˜f∗ (w ˜ ) = minX∈Φf e∈X w ˜e (w and δ˜f (w w) ˜ − F˜ ∗ (w ˜ ) and δ˜f (w ˜ ) are now gradual numbers and they are computed by means of δ˜X (w operations of the sum, the subtraction and the minimum in the space of gradual numbers. Suppose now that for every weight we , e ∈ E, there is given a fuzzy interval ˜ e = [w W ˜e− , w ˜e+ ], where w ˜e− is a gradual lower bound and w ˜e+ is a gradual upper ˜ bound of We . Now the fuzzy deviations are fuzzy intervals which can be also − ˜+ described by pairs of gradual numbers, namely ∆˜X = [δ˜X , δX ] and ∆˜f = [δ˜f− , δ˜f+ ]. In order to apply the interval methods, given in Section 3, to the fuzzy interval computations, we need to extend extreme configurations to the fuzzy extreme ones. Let A ⊆ E be a given subset of elements. In the fuzzy extreme configuration ˜+ w ˜e+ and all the remaining ones have A all elements e ∈ A have gradual weights w − ˜− gradual weights w ˜e . Similarly, in the fuzzy fuzzy extreme configuration w A all − elements e ∈ A have gradual weights w ˜e and all the remaining ones have gradual ˜+ ˜− weights w ˜e+ . Now w ˜e (w ˜e (w A ) and w A ) are gradual weights in fuzzy extreme + − ˜ A and w ˜ A , respectively. A fuzzy counterpart of Proposition 1 is configurations w the following one: Proposition 2. The following equalities hold: − = δ˜X
δ˜f− =
min
˜+ ˜ ) = δ˜X (w ˜− δ˜X (w X ), δX =
min −
˜ ), δ˜f+ = δ˜f (w
w ˜ ∈×e∈E {w ˜e− ,w ˜e+ } ˜ ∈×e∈E {w w ˜ e ,w ˜e+ }
max
max
w ˜ ∈×e∈E {w ˜e− ,w ˜e+ }
˜ ∈×e∈E {w w ˜e− ,w ˜e+ }
˜ ) = δ˜X (w ˜+ δ˜X (w X ),
˜ ), δ˜f (w
Proof. The proof that the minimum and maximum are attained in fuzzy extreme configurations follows from [8] and the fact that functions δX and δf are locally ˜− ˜+ monotonic with respect to each variable. The proof that w X (w X ) minimizes ˜ ) follows from the existence of a link between the interval (maximizes) δ˜X (w ˜ e = [w model and the fuzzy one. Observe that W ˜e− , w ˜e+ ] = [Aw˜e− (λ), Aw˜e+ (λ)] for λ ∈ [0, 1]. Hence, from Proposition 1 we conclude that for every fixed λ the maximum and minimum are attained either at Aw˜e− (λ) or Aw˜e+ (λ). ⊓ ⊔ Notice that, similarly to the interval-valued case, we can identify the fuzzy extreme configurations that minimize and maximize the fuzzy deviation for a solution but it may be a hard task for an element. ˜X From Proposition 2, we immediately obtain algorithms for Computing ∆ determining the gradual lower and upper bounds of fuzzy interval ∆˜X for a given solution X ∈ Φ. Algorithm 1 is the one for determining the gradual lower − + bound δ˜X (an algorithm for the gradual upper bound δ˜X is similar). A key ˜− line in Algorithm 1 is line 4, in which we compute the value of F˜ ∗ (w X ). This is a gradual number expressing the fuzzy value of the weight of an optimal
˜X Algorithm 1: Determining the gradual lower bound of ∆ ˜ e = [w Input: Solution X ∈ Φ, fuzzy weights W ˜e− , w ˜e+ ], e ∈ E. − ˜X . Output: The gradual lower bound δ˜X of ∆ 1 foreach e ∈ E do ˜− ˜− 2 if e ∈ X then w ˜e (w ˜e− else w ˜e (w ˜e+ X) ← w X) ← w P ˜ (X, w ˜− ˜− 3 F ˜e (w X) ← X ) /*the sum of gradual numbers e∈X w P − ∗ ˜ ˜− ˜− ˜ X ) by solving minX∈Φ F˜ (X, w 4 Compute F (w ˜e (w X) X ) = minX∈Φ e∈X w − − − ∗ ˜ ˜ ˜ ˜ X ) − F (w ˜ X ) /*the subtraction of gradual numbers 5 δX ← F (X, w ˜− 6 return δ
*/ */
X
˜∗ ˜ − ) ˜− solution in fuzzy configuration w X . From the technical point of view, F (w X is a function from [0, 1] to IR and it can be obtained by solving a parametric version of the combinatorial optimization problem P. Recall that in a parametric problem every element weight is specified as a function w(λ), λ ∈ IR, and we wish to compute function F ∗ (λ), so that F ∗ (λ) is the weight of an optimal solution if the element weights are set to w(λ). In our case the weights are given as the assignment functions Aw˜e (w˜ − ) (λ) for all e ∈ E. In particular, if we apply X fuzzy intervals of the L-R type, then we can use assignment functions of the ˜− form (6). In order to compute the value of F˜ ∗ (w X ) some known methods for solving parametric problems with linearly varying weights can be applied (see e.g. [6, 12, 13]). These algorithms can be directly applied if the fuzzy intervals are trapezoidal or triangular ones (their shape functions are linear). They can be also applied if the fuzzy intervals are of the L-L type (their right and left shape functions are the same) since function L can be then easily linearized. From the knowledge of fuzzy deviation ∆˜X , we can also obtain the degrees of possibility and necessity that a given solution X is optimal (see (7) and (8)). Let us illustrate our algorithms by an example. Consider Shortest Path problem shown in Fig. 2. The arc weights are given as triangular fuzzy intervals: ˜ a = (1, 1, 1). ˜ a = (2, 1, 4), W ˜ a = (6, 2, 1), W ˜ a = (2, 1, 4), W ˜ a = (2, 2, 2), W W 5 4 3 2 1 We wish to compute the fuzzy deviation for the path composed of arcs a1 ˜− and a3 , that is ∆˜{a1 ,a3 } . In Fig. 2a the fuzzy configuration w {a1 ,a3 } and in + w ˜ Fig. 2b the fuzzy configuration {a1 ,a3 } are shown. Observe, that these configurations induce two parametric shortest path problems in which the parametric weights are computed by means of formula (6). Solving the parametric prob˜∗ ˜ + ˜− lems we obtain the gradual numbers F˜ ∗ (w {a1 ,a3 } ) and F (w {a1 ,a3 } ). Comparing − + ˜ {a1 ,a3 } ) and F˜ ({a1 , a3 }, w ˜ {a1 ,a3 } ) we obtain the gradual them to F˜ ({a1 , a3 }, w ˜ lower and the gradual upper bounds of ∆{a1 ,a3 } (see Fig. 2c). We can see that Π({a1 , a3 } is optimal) = 4/7 and N({a1 , a3 } is optimal) = 0. We can obtain also an additional information, for instance Π(δ{a1 ,a3 } ≤ 1 35 ) = 4/5. ˜f for some practical problems Contrary to the optimality of Computing ∆ solutions, there is no an easy characterization of the optimality of the elements
a)
b) 6 − 2(1 − λ)
2 − 2(1 − λ) a1
s
a3
a2
s
t
a5 1 − (1 − λ) a2
a4 2 − (1 − λ)
2 + 4(1 − λ)
c)
a3
a1
a5 1 + (1 − λ)
2 + 4(1 − λ)
6 + (1 − λ)
2 + 2(1 − λ)
t
a4 2 − (1 − λ)
˜{a ,a } ∆ 1 3
1 4 5 4 7
13 5
4
9
Fig. 2. An example of fuzzy-valued Shortest Path Problem.
and the computational complexity of determining ∆˜f for a given element f ∈ E strongly depends on a particular problem P. ˜ ) = F˜f∗ (w ˜ ) and we wish to determine δ˜f− and δ˜f+ , that Recall that δ˜f (w w)− ˜ F˜ ∗ (w ˜ ). While computing is the gradual numbers that minimize and maximize δ˜f (w these gradual bounds for a given element f ∈ E two problems arises. The first ˜ ). one is computing the extreme configurations that minimize and maximize δ˜f (w It turns out that the problem of computing the proper extreme configurations is strongly NP-hard for some well-known, polynomially solvable problems like Shortest Path, Minimum Assignment and Minimum Cut. These problems remain strongly NP-hard even in the interval-valued case. The second problem is computing a gradual minimum over all solutions that contain element f , that is the value of F˜f∗ (w w). ˜ In this section we briefly present some particular problems for which ∆˜f can be efficiently determined. The first problem is Minimum Spanning Tree. Given is an undirected graph G = (V, E) with edge weights specified as fuzzy intervals. Set Φ consists of all spanning trees of G. We wish to determine interval ∆˜f (more precisely the possibility distribution µ∆˜f (x)) for a given edge f ∈ E. The following proposition immediately follows from the result obtained in [9]: ˜− Proposition 3. The fuzzy configuration w {f } minimizes and the fuzzy configu+ ˜ {f } maximizes gradual deviation δ˜f (w ration w w). ˜ Applying the parametric approach proposed in [6] we can compute the values of ˜ ∗ ˜ + ). Applying a slightly modified parametric approach we ˜− F˜ ∗ (w {f } {f } ) and F (w ˜ ∗ ˜ + ). In consequence we obtain δ˜− and δ˜+ ˜− can also compute F˜f∗ (w f f {f } {f } ) and Ff (w for the problem. The second problem for which ∆˜f can be efficiently computed is Shortest Path when the input graph is restricted to be edge series-parallel digraph
(see [14] for a description of this class of graphs). In this problem there is given an edge series parallel digraph G = (V, A) with two distinguished nodes s and t and arc weights specified as fuzzy intervals. Set Φ consists of all paths from s to t in G. Denote by P red(f ) the set of all arcs that precede arc f and by Succ(f ) the set of all arcs that succeed arc f on a path from s to t in G. The following proposition is a consequence of the results from [3]: ˜− Proposition 4. The fuzzy configuration w {P red(f )∪{f }∪Succ(f )} minimizes and + ˜ {P red(f )∪{f }∪Succ(f )} ) maximizes gradual deviation δ˜f (w w). ˜ the fuzzy configuration w Applying a parametric approach proposed in [12] we can compute the value of ˜ ), where w ˜ is one of the two fuzzy configurations specified in Proposition 4. F˜ ∗ (w ˜ ) for arc f = (k, l) can be computed using the fact that The value of F˜f∗ (w ∗ ∗ ∗ ˜ ) = F˜s−k ˜) + w ˜ ) + F˜l−t ˜ ), where F˜s−k ˜ ) is the gradual weight of a F˜f∗ (w (w ˜f (w (w (w ∗ ˜ ˜ ) is the gradual weight of a shortest path shortest path from s to k and Fl−t (w from l to t in configuration w w. ˜ Both gradual numbers can be also computed by means of the parametric approach proposed in [12].
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