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Fractional multi-commodity flow problem: Duality and optimality conditions
Applied Mathematical Modelling 38 (2014) 2151–2162
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Fractional multi-commodity flow problem: Duality and optimality conditions q Ashkan Fakhri, Mehdi Ghatee ⇑ Department of Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran Intelligent Transportation Systems Research Institute, Amirkabir University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 24 June 2012 Received in revised form 31 August 2013 Accepted 8 October 2013 Available online 29 October 2013 Keywords: Combinatorial optimization Fractional programming Duality Strong complementary slackness Multi-commodity flow problem
a b s t r a c t This paper deals with multi-commodity flow problem with fractional objective function. The optimality conditions and the duality concepts of this problem are given. For this aim, the fractional linear programming formulation of this problem is considered and the weak duality, the strong direct duality and the weak complementary slackness theorems are proved applying the traditional duality theory of linear programming problems which is different from same results in Chadha and Chadha (2007) [1]. In addition, a strong (strict) complementary slackness theorem is derived which is firstly presented based on the best of our knowledge. These theorems are transformed in order to find the new reduced costs for fractional multi-commodity flow problem. These parameters can be used to construct some algorithms for considered multi-commodity flow problem in a direct manner. Throughout the paper, the boundedness of the primal feasible set is reduced to a weaker assumption about solvability of primal problem which is another contribution of this paper. Finally, a real world application of the fractional multi-commodity flow problem is presented. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction In many application contexts, several physical commodities, vehicles, or messages, each governed by their own network flow constraints, share the same network [2,3]. Multi-commodity flow problem is achieved in a lot of network design and transportation problems [4]. In this problem some commodities from special origins should be transmitted to some other destinations with a minimum total cost [2]. Solving this problem under integer restriction is NP-hard [2]. When multiple objective functions or uncertain ones are given, the solution process of multi-commodity problem is harder than traditional methods [5]. When only two objective functions are considered into account e.g., cost minimization and reliability maximization, one can optimize the ratio of these objective functions. The provided fractional programming problem can be solved applying some famous approaches. Besides, fractional programs are happened in a lot of practical problems [6,7] which can be pursued in other network analysis. For fractional multi-commodity problems to satisfy the demand for each commodity at each node without violating the constraints imposed by the supply–demand and capacity, one can consider the following
q
This paper was partially supported by Intelligent Transportation Systems Research Institute, Amirkabir University of Technology, Tehran, Iran.
⇑ Corresponding author at: Department of Computer Science, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran. Tel.: +98 21 64542510; fax: +98 21 66497930. E-mail addresses: [email protected] (A. Fakhri), [email protected] (M. Ghatee). URL: http://www.aut.ac.ir/ghatee (M. Ghatee). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.10.032
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maximum fractional multi-commodity flow model in which G ¼ ðN; AÞ is a network with N and A as the sets of n nodes and m links:
PK P k¼1
maximize f ðxÞ ¼ PK P k¼1
k k ði;jÞ2A c i;j xi;j k k ði;jÞ2A di;j xi;j
þa þb
ð1:1Þ
;
8P P k > xk j:ðj;iÞ2A xkj;i ¼ bi ; 8i 2 N; 8k ¼ 1; . . . ; K; > < Pj:ði;jÞ2A i;j K k s:t: 8ði; jÞ 2 A; k¼1 xi;j 6 ui;j ; > > : k xi;j P 0; 8ði; jÞ 2 A; 8k ¼ 1; . . . ; K:
ð1:2Þ
Here, xki;j is a nonnegative variable regarding to the amount of flow of kth commodity which streams through link ði; jÞ; ui;j is k the capacity of link ði; jÞ and K is the number of commodities. Moreover, for commodity k; cki;j and di;j are the unit-reliability k and the unit-cost of flow through link ði; jÞ, respectively, and bi is the supply or demand of node i, defined by positive or negative numbers. In this model a and b are two positive constants regarding two lower levels for the corresponding objective functions. Applying vector and matrix notations, this model can be rewritten as the following maximum fractional linear programming model: t
a maximize f ðxÞ ¼ dc t xþ xþb
ð1:3Þ
s: t: x 2 S1 ¼ fx : Ax 6 b; x P 0g: Here A is a ð2:n:K þ mÞ by ðm:KÞ matrix as follows:
k¼1
k¼2
k ¼ K
ði; jÞ 2 A ði; jÞ 2 A 2
0
N
6 N 6 6 6 0 6 6 0 6 A¼6 6 .. 6 . 6 6 0 6 6 4 0 I
ði; jÞ 2 A
; 0
0 N
0
0 0
N .. .
0 .. .
.. .
0 .. .
0
N
0
N
I
I
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
8i 2 N 8i 2 N 8i 2 N 8i 2 N .. .
;
ð1:4Þ
8i 2 N 8i 2 N 8ði; jÞ 2 A
in which N is the n by m incidence matrix of the network G and I is the m by m identity matrix. Also b is a column vector with 2:n:K þ m components as follows: 1 3 b 6 17 6 b 7 6 2 7 6 b 7 7 6 6 27 6 b 7 7 b¼6 6 .. 7 6 . 7 7 6 6 bK 7 7 6 6 K7 4 b 5
2
k
k¼1 k¼2 k¼2 ; .. .
ð1:5Þ
k¼K k¼K ði; jÞ 2 A
u k
k¼1
k
k t
where b ¼ ½b1 ; b2 ; . . . ; bn is supply–demand vector with respect to commodity k, for k ¼ 1; . . . ; K, and x; c and d are column k vectors with m:K components including the values of xki;j ; cki;j and di;j , respectively. Also, it is assumed that S1 is nonempty and t d x þ b > 0 for all x in S1 . In addition, we assume that the optimal solution of fractional multi-commodity flow problem (1.1) and (1.2) is bounded. Note that the objective function f ðxÞ is a pseudo-linear (both pseudo-convex and pseudo-concave) function and any local optimal solution of problem (1.3) is also a global optimal solution [8]. For this problem, Chadha and Chadha [1] proposed a linear dual problem and investigated weak duality, strong direct duality and weak complementary slackness theorems (see [9], also). In [10] a network simplex algorithm and also a similar dual problem is proposed for this problem with a single commodity namely a fractional minimal cost flow problem. In continuation, Sherali [11] showed that the resulted theorems
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and duality introduced in [10] can be directly found by linear programming transformations. This paper follows the Sherali’s approach for obtaining same results for multi-commodity flow problem. It is worthwhile to mention that in [1,10] there is no reference to the strong complementary slackness theorem. In fact, a direct proof for this theorem seems to be more complicated than the ones for the other duality relations. For treating this difficulty, we show in Section 2, via some straightforward results, that the dual problem proposed by Chadha and Chadha [1] can be derived by duality theory of linear programming. After presenting the new proofs for duality relations, we can prove indirectly the strong complementary slackness theorem applying the traditional duality principles in linear programming when well-known Charnes and Cooper [12] variable transformation is done on problem (1.3). Since the results of this paper are derived only by solvability assumption on the primal problem (1.3), the proposed theorems are more applicable than previous works based on boundedness of S1 . In fact, our assumption is weaker because if S1 is bounded then the problem (1.3) is reduced to a problem in which a continuous function should be maximized over a compact set, which certainly is a solvable problem (see, e.g., [8]). Based on these new results on fractional linear programming, the dual problem of fractional multi-commodity flow problem is presented in Section 3 and a new reduced cost is defined to check optimality of solutions. Finally the complementary slackness conditions are proved and a real world application is introduced for fractional multi-commodity flow problem. Section 4 ends this paper with some brief remarks. 2. Duality properties of fractional linear programming problem By the Charnes–Cooper variable transformation [12], t
1
t ¼ ðd x þ bÞ ;
t
1
w ¼ ðd x þ bÞ x;
x 2 S1 ;
one can rewrite the problem (1.3) as the following problem:
maximize hðw; tÞ ¼ ct w þ at; s:t:
n o t ðw; tÞ 2 S2 ¼ ðw; tÞ : Aw bt 6 0; d w þ bt ¼ 1; w P 0; t > 0 :
ð2:1Þ
After replacing the last constraint with t P 0, the following linear programming problem can be achieved:
maximize hðw; tÞ ¼ ct w þ at; s: t:
n o t ðw; tÞ 2 S3 ¼ ðw; tÞ : Aw bt 6 0; d w þ bt ¼ 1; w P 0; t P 0 :
ð2:2Þ
The following two lemmas indicate some relationships among problems (1.3), (2.1) and (2.2). Lemma 2.1 (see [13]). The problem (1.3) has optimal solution if and only if the problem (2.1) has one. In fact if x is an optimal 1 1 t t solution of problem (1.3) then ðw; tÞ ¼ ððd x þ bÞ x; ðd x þ bÞ Þ is an optimal solution of problem (2.1), and if ðw; tÞ is an optimal solution of problem (2.1) then x ¼ ðw=tÞ is an optimal solution of problem (1.3). Obviously, in the both cases optimal values are equal.
Lemma 2.2. If ðw ; t Þ solves problem (2.1) then ðw ; t Þ solves problem (2.2) too. Conversely, if ðw ; t Þ solves problem (2.2) and t > 0 then ðw ; t Þ solves problem (2.1) too. ^ ^tÞ 2 S3 such Proof. Assume that ðw ; t Þ solves problem (2.1). In contradiction to desired result, suppose that there exists ðw; that
^ þ a^t > ct w þ at ct w
^ ^tÞ > hðw ; t ÞÞ: ði:e: hðw;
ð2:3Þ
For each k 2 ð0; 1Þ, we have
^ ^tÞ þ ð1 kÞðw ; t Þ 2 S2 ðwk ; t k Þ :¼ kðw;
ð2:4Þ
^ þ a^t ct w at Þ þ ct w þ at > ct w þ at ¼ hðw ; t Þ; hðwk ; tk Þ ¼ ct wk þ at k ¼ kðct w
ð2:5Þ
and
where the last inequality of (2.5) is valid because of (2.3). But the combination of (2.4) and (2.5) contradicts optimality of ðw ; t Þ for problem (2.1). Hence ðw ; t Þ solves problem (2.2) too. Conversely, if ðw ; t Þ solves problem (2.2) and t > 0 then ðw ; t Þ solves problem (2.1) because objective functions of the problems (2.1) and (2.2) are similar, S2 # S3 and ðw ; t Þ 2 S2 . h The following corollary is directly deduced from Lemmas 2.1 and 2.2:
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t
t
1
Corollary 2.1. If x solves problem (1.3) then ðw ; t Þ ¼ ððd x þ bÞ x ; ðd x þ bÞ Þ solves problem (2.2) and f ðx Þ ¼ hðw ; t Þ. Conversely, if ðw ; t Þ solves problem (2.2) and t > 0 then x ¼ ðw =t Þ solves problem (1.3) and f ðx Þ ¼ hðw ; t Þ. Now consider the linear dual of the problem (2.2) (see, e.g., [14]):
minimize gðy; zÞ ¼ z; s:t:
n o t ðy; zÞ 2 S4 ¼ ðy; zÞ : At y þ dz P c; b y þ bz P a; y P 0 :
ð2:6Þ
t
After replacing the inequality b y þ bz P a with equality, the following problem can be derived from problem (2.6):
minimize gðy; zÞ ¼ z s:t:
n o t ðy; zÞ 2 S5 ¼ ðy; zÞ : At y þ dz P c; b y þ bz ¼ a; y P 0 :
ð2:7Þ
which is exactly similar to the dual form proposed by Chadha and Chadha [1]. Lemma 2.3 reveals some relationships among optimal values of the preceding problems. Lemma 2.3. Assume that problem (1.3) is solvable. Then (i) Problem (2.6) is solvable and its optimal value is equal to optimal value of problem (2.2) (and hence equal to optimal value of problem (1.3), regarding Corollary 2.1. ; zÞ of problem (2.6) is a feasible solution of problem (2.7). (ii) Any optimal solution ðy ^; ^zÞ is optimal for problem (2.6) if and only if it is optimal for problem (2.7). (iii) ðy
^ ^tÞ that solves problem Proof. ðiÞ Let ^ x be an optimal solution of problem (1.3). Then regarding Corollary 2.1, there exists ðw; (2.2) and ^t > 0. Hence according to the strong duality theorem of linear programming [14], problem (2.6) has an optimal solution with the same optimal value of problem (2.2). ; zÞ is optimal for problem (2.6). According to Corollary 2.1, problem (2.2) has an optimal solution ðiiÞ Assume that ðy ðw ; t Þ so that t > 0. Therefore, applying the weak complementary slackness theorem of linear programming [14], we have t þ bz ¼ a. This completes the proof of part ðiiÞ. b y ^; ^zÞ is optimal for problem (2.6). Then as a result of the part ðiiÞ; ðy ^; ^zÞ is feasible for problem (2.7) and ðiiiÞ Assume that ðy hence optimal for it (because objective functions of problems (2.6) and (2.7) are similar and S5 # S4 ). Conversely, assume that ^; ^zÞ is optimal for problem (2.7). Obviously ðy ^; ^zÞ is feasible for problem (2.6). In cotradiction to desired result, suppose that ðy ^; ^zÞ is not optimal for problem (2.6). Then there exists an optimal solution ðy ; zÞ of problem (2.6) such that z < ^z (existence ðy ; zÞ is feasible for problem (2.7) too (see of optimal solution for problem (2.6) is guaranteed by the part (i)). But in this case, ðy ^; ^zÞ for problem (2.7). This contradiction completes the proof. h part ðiiÞ), which contradicts optimality of ðy In the sequel, we prove duality theorems. Note that Theorems 2.1, 2.2 and 2.3 have been proved by Chadha and Chadha [1] with another approach. However, in what follows, these theorems are proved applying the traditional duality theorems for linear programming problems. In addition, in Theorems 2.4 and 2.5, strong complementary slackness property is stated which is the first time based on the best of our knowledge. Also in Theorem 3.1 the optimality conditions for fractional multicommodity flow problem are presented. Theorem 2.1 (Weak Duality Theorem [1]). For each x in S1 and for each ðy; zÞ in S5 , we have f ðxÞ 6 gðy; zÞ. t
1
t
1
Proof. Let x belongs to S1 . Then ðw; tÞ ¼ ððd x þ bÞ x; ðd x þ bÞ Þ is in S3 and f ðxÞ ¼ hðw; tÞ. Regarding weak duality theorem of linear programming (see, e.g., [14]):
hðw; tÞ 6 z 8ðy; zÞ 2 S4 : But S5 S4 and hence
f ðxÞ ¼ hðw; tÞ 6 z ¼ gðy; zÞ 8ðy; zÞ 2 S5 : This completes the proof. h Next theorem is strong direct duality theorem which is a direct corollary of Lemma 2.3. Theorem 2.2 (Strong Direct Duality Theorem [1]). If problem (1.3) if solvable then problem (2.7) is solvable and their optimal values are equal.
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Proof. See Lemma 2.3. h The following example illustrates that if problem (2.7) is solvable then problem (1.3) is not necessarily solvable. Example 2.1. Consider the following problem:
maximize
x1 x1 þx2
s:t:
ð2:8Þ
x1 P 1; x2 P 0: Then, ðy ; z Þt ¼ ð0; 0Þt solves the dual of (2.8), but (2.8) has not optimal solution.
Theorem 2.3 (Weak Complementary Slackness Theorem [1]). Let u and v be slack and surplus column vectors associated with ^ Þ and ðy ^; ^z; v ^ Þ be feasible x; u problems (1.3) and (2.7), respectively, and assume that problem (1.3) is solvable. Furthermore, let ð^ ^ Þ solves problem (1.3) and ðy ^; ^z; v ^ Þ solves problem (2.7) if and solutions of problems (1.3) and (2.7), respectively. Then, ð^ x; u only if
v^ t ^x þ u^t y^ ¼ 0: ^; u ^ Þ solves problem (1.3) and ðy ^; ^z; v ^ Þ solves problem (2.7). According to Corollary 2.1, Proof. First assume ðx 1 1 t t ^; ^z; v ^ Þ solves ðw ; t Þ ¼ ððd ^ x þ bÞ ^ x; ðd ^ x þ bÞ Þ solves problem (2.2). On the other hand, due to the part (iii) of Lemma 2.3, ðy problem (2.6). Hence applying the weak complementary slackness theorem of linear programming [14] one can state: t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ w ¼ 0: ðAw bt Þ y
ð2:9Þ
Dividing the Eq. (2.9) by t > 0, we have t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ ^x ¼ 0: ðA^x bÞ y
ð2:10Þ
Therefore
^t y ^ ¼ 0 and u
v^ t ^x ¼ 0
ð2:11Þ
and hence
^t y ^ þ v^ t ^x ¼ 0: u
ð2:12Þ
^; u ^ and v ^ , we have Conversely, assume that u y þ v x ¼ 0. Because of nonnegativity of ^ x; y ^t ^
^t y ^ ¼ 0 and u
^t^
v^ t ^x ¼ 0
ð2:13Þ
and hence t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ ^x ¼ 0: ðA^x bÞ y
ð2:14Þ t
Multiplying the above equations by positive t ¼ 1=ðd ^ x þ bÞ and defining w :¼ t ^ x, we have
ðw ; t Þ 2 S3 ;
^; ^zÞ 2 S4 ; ðy
ð2:15Þ t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ w ¼ 0: ðAw bt Þ y
ð2:16Þ
Now with respect to the weak complementary slackness theorem of linear programming [14], ðw ; t Þ solves problem (2.2) ^; ^zÞ solves problem (2.6). Therefore ^ ^; ^zÞ solves problem (2.7) (see and ðy x solves problem (1.3) (see Corollary 2.1) and and ðy part (iii) of Lemma 2.3). This completes the proof. h ^ Þ solves problem (1.3) and ðy ^; ^z; v ^ Þ solves problem (2.7) if and Remark 2.1. By the same assumptions of Theorem 2.3, ð^ x; u only if
v^ a :^xa ¼ 0; ^ b :y ^b ¼ 0; u where a and b are two arbitrary indices for the corresponding vectors. The following theorem is one of the main results of this paper. Note that we have not supposed the feasible set S1 to be bounded, see [9].
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Theorem 2.4 (Strong Complementary Slackness Theorem). Assume that problem (1.3) is solvable, and let u and v be slack and ^ Þ and ðy ^; ^z; v ^Þ surplus column vectors associated with problems (1.3) and (2.7), respectively. Then there exist optimal solutions ð^ x; u for problems (1.3) and (2.7), respectively, such that
^þy ^ > 0 and u
v^ þ ^x > 0:
ð2:17Þ
Proof. Regarding Corollary 2.1, problem (2.2) is solvable. Now with respect to the strong complementary slackness theorem ^; ^zÞ of problems (2.2) and (2.6), respectively, such that of linear programming [14], there exist optimal solutions ðw ; t Þ and ðy
^ þ ðbt Aw Þ > 0; y
ð2:18Þ
^ þ d^z cÞ > 0; w þ ðAt y
ð2:19Þ
t ^ þ b^z aÞ > 0: t þ ðb y
ð2:20Þ t
^; ^zÞ is an optimal solution of problem (2.7). Hence b y ^ þ b^z a ¼ 0, and relation According to part (iii) of Lemma 2.3, ðy (2.20) can be rewritten as follows:
t > 0:
ð2:21Þ t
^ :¼ A y ^ þ d^z c, we have ^ Dividing the relations (2.18) and (2.19) by t and defining ^ x :¼ w =t and v x 2 S1 and
^=t Þ þ ðb A^xÞ > 0 and ^x þ ðv^ =t Þ > 0; ðy ^ :¼ ðb A^ in which ^ x is an optimal solution of problem (1.3) (see Corollary 2.1. Now defining u xÞ, we have
^=t Þ þ u ^ > 0 and ^x þ ðv^ =t Þ > 0; ðy
ð2:22Þ
or equivalently
^þu ^ > 0 and ^x þ v^ > 0; y
ð2:23Þ
^; u ^ and v ^ . Therefore we have found soluwhere (2.23) is a direct corollary of (2.22), positivity of t and nonnegativity of ^ x; y ^ Þ and ðy ^; ^z; v ^ Þ as optimal solutions for problems (1.3) and (2.7), respectively, such that ^ ^; y ^ and v ^ satisfy (2.17). tions ð^ x; u x; u This completes the proof. h
Remark 2.2. Solvability of problem (1.3) is crucial, because as we illustrated in Example 2.1, if problem (2.7) is solvable then problem (1.3) is not necessarily solvable. Note that replacing the unrestricted variable z by two nonnegative variables zþ and z , problem (2.7) can be rewritten as:
minimize zþ z s: t:
ð2:24Þ þ
At y þ dz dz P c;
t
b y þ bzþ bz ¼ a;
y; zþ ; z P 0:
Quite similar to Theorem 2.4, we can state the following result: Theorem 2.5. Assume that problem (1.3) is solvable, and let u and v be slack and surplus column vectors associated with problems ^ Þ and ðy ^; z^þ ; z^ ; v ^ Þ for problems (1.3) and (2.24), respectively, (1.3) and (2.24), respectively. Then, there exist optimal solutions ð^ x; u such that
^þy ^ > 0; u
v^ þ ^x > 0
and z^þ ; z^ > 0:
ð2:25Þ
3. Optimality conditions for fractional multi-commodity flow problem In this section, we consider the fractional multi-commodity flow problem (1.1) subjected to (1.2) and try to find some optimality conditions for this problem. Doing so permits us to assess whether or not we have found an optimal solution to the problem. It is also permits us to interpret various algorithms as particular methods for solving the optimality conditions, and in several instances even suggests novel algorithmic approaches for solving the problem we were studying [2]. Since the fractional multi-commodity flow problem is a fractional linear program, based on the duality theory of linear programming, we can present the dual of the fractional multi-commodity flow problem (1.1) and (1.2). Because of different bundle constraints for every link ði; jÞ, mass balance constraints for each node-commodity combination and an auxiliary constraint, the dual of this problem has three types of variables: a price hi;j on each link ði; jÞ, node potential pki for each combination of commodity k and node i and also an auxiliary variable z related to the auxiliary constraint. Applying these dual p variables, it is possible to define the reduced cost ck; i;j of link ði; jÞ with respect to the commodity k as follows:
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p k k k ck; i;j :¼ ðc i;j di;j z hi;j Þ pi þ pj ; k
ð3:1Þ
where
z¼
! K X X X 1 k aþ bi pki þ ui;j hi;j : b ði;jÞ2A k¼1 i2N
ð3:2Þ
The following theorem states that a primal feasible solution xki;j and a dual feasible solution ðhi;j ; z; pki Þ are optimal to the respective problems if they satisfy a new set of complementary slackness conditions. Theorem 3.1 (Fractional Multi-commodity Flow Complementary Slackness Conditions). Assume that the fractional multicommodity flow problem (1.1) subjected to (1.2) is solvable. The commodity flows xki;j are optimal for this problem if and only if they are feasible and for some nonnegative link prices hi;j , (unrestricted is sign) node potentials pki and scalar z, the following complementary slackness conditions are satisfied:
! K X k xi;j ui;j hi;j ¼ 0; for all links ði; jÞ 2 A;
ð3:3Þ
k¼1
p xki;j ck; i;j ¼ 0; for all links ði; jÞ 2 A and all k ¼ 1; . . . ; K;
ð3:4Þ
p ck; i;j 6 0; for all links ði; jÞ 2 A and all k ¼ 1; . . . ; K;
ð3:5Þ
in which,
p ck; i;j
and z are defined by Eqs. (3.1) and (3.2), respectively.
Proof. To prove this theorem, we consider matrix form of fractional multi-commodity flow problem represented in (1.3). Noting to Eqs. (1.4) and (1.5), the nonnegative dual variables piþ;k and p;k can be defined for a couple of mass balance coni straints for each node-commodity combination ði; kÞ and also the nonnegative variables hi;j with respect to the capacity constraint on link ði; jÞ. According to problem (2.7), the dual problem of fractional multi-commodity flow problem can be stated as follows:
minimize gðh; z; pÞ ¼ z s: t: k
þ;k k ðpþ;k p;k p;k 8k ¼ 1; . . . ; K; 8ði; jÞ 2 A; i i Þ ðpj j Þ þ hi;j þ di;j z P c i;j ; ! K X X k X k ;k X bi pþ;k þ bi pi ui;j hi;j þ bz ¼ a; i k¼1
i2N
i2N
ð3:6Þ
ði;jÞ2A
;k pþ;k P 0; 8i 2 N; i ; pi
hi;j P 0;
8ði; jÞ 2 A:
Applying a variable transformation in which the unrestricted variable pki is used instead of pþ;k p;k i i , the dual of fractional multi-commodity flow problem can be stated as follows:
minimize gðh; z; pÞ ¼ z s: t:
pki pkj þ hi;j þ dki;j z P cki;j ; 8k ¼ 1; . . . ; K; 8ði; jÞ 2 A;
ð3:7Þ
K X X X k bi pki ui;j hi;j þ bz ¼ a; k¼1 i2N
hi;j P 0;
ði;jÞ2A
8ði; jÞ 2 A:
Now, note that the desired result is a straightforward corollary of Theorems 2.2 and 2.3.
h
Theorem 3.2. Let the commodity flows xki;j be optimal in the fractional multi-commodity flow problem (1.1) and (1.2), hi;j be the optimal link prices and pki be optimal node potentials with respect to commodity k. Then for each commodity k, the flow variables xki;j solve the following (uncapacitated) fractional maximum cost flow problem:
P ck xk þa max Pði;jÞ2A i;jk i;jk
d x þb ði;jÞ2A i;j i;j
s:t:
ð3:8Þ N xk ¼ b k ; xki;j P 0; for all ði; jÞ 2 A;
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are defined as the following: in which, cki;j and a
cki;j ¼ cki;j hi;j and
a ¼ a þ
K X X
k0 b i2N i
k0 ¼1 k0 –k
0
pki þ
X
u h ði;jÞ2A i;j i;j
Proof. The structure of proof is similar to [2]. Since xki;j are optimal flows, hi;j are optimal link prices and pki are optimal node potentials for the fractional multi-commodity flow problem, these variables together with some z, defined by Eq. (3.2), satisfy the complementary slackness conditions (3.3)–(3.5). But (3.4) and (3.5) are the optimality conditions for the fractional . h maximum cost flow problem for commodity k with link costs cki;j and numerator constant a Attempting to find a sequential method to solve the fractional multi-commodity flow problem is postponed to our future works. Finally, in what follows, we present a real world example of problem (1.1). This example is only one of the several applications of the presented fractional multi-commodity flow problem. 4. Tourism company application A tourism company wishes to select appropriate and reasonable pathways to transfer its clients (tourists) from some origins to some destinations in northern cities of Iran near to Caspian see depicted in Fig. 1. There are different kinds of nodes: A is an origin of tourists, C is both origin and destination, E; F; I and K are destinations and the others are middle cities. Consider the network of Fig. 2, whose nodes are origin and destination cities of Fig. 1 and the links are corresponding to the real paths between each couple of nodes. In this example, the following terminologies and notations are used. A group of tourists who are conveyed between a certain origin p and a certain destination q, are considered as commodity p}q, and the set of such tourism groups (commodities) is denoted with K:
K ¼ fA}C; A}E; A}F; A}I; A}K; C}E; C}F; C}I; C}Kg: p}q
For each group of tourists p}q; b p}q bp
¼
p}q bq
¼b
is the number of travelers from p to q; hence for each p}q 2 K we set
p}q
and p}q
bi
cp}q i;j p}q di;j
¼ 0;
and
p}q di;j
8i – p; q: are unit-reliability and unit-cost values to travel the tourism group p}q through link ði; jÞ. It is assumed that
¼ di;j (see the link labels illustrated in Fig. 2) and ui;j ¼ 8 for each link ði; jÞ.
Fig. 1. Northern cities of Iran.
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Fig. 2. The imaginary network with respect to the topological network of Fig. 1.
Table 1 The number of travelers between different origins and destinations. Tourism groups p}q b
p}q
A}C
A}E
A}F
A}I
A}K
C}E
C}F
C}I
C}K
2
3
2
5
3
1
1
3
2
Table 2 Degrees of reliability cp}q for link ði; jÞ and tourism group p}q. i;j Links
ðA; BÞ ðA; FÞ ðA; GÞ ðA; JÞ ðA; LÞ ðB; AÞ ðB; CÞ ðB; EÞ ðC; BÞ ðC; DÞ ðD; CÞ ðD; EÞ ðE; BÞ ðE; DÞ ðE; FÞ ðF; AÞ ðF; EÞ ðF; GÞ ðG; AÞ ðG; FÞ ðG; HÞ ðH; GÞ ðH; IÞ ðH; JÞ ðI; HÞ ðJ; AÞ ðJ; HÞ ðJ; KÞ ðK; JÞ ðK; LÞ ðL; AÞ ðL; KÞ
Tourism Groups A}C
A}E
A}F
A}I
A}K
C}E
C}F
C}I
C}K
0.85 0.6 -M -M -M -M 0.85 0.85 -M -M 0.8 -M -M 0.8 -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
0.85 0.6 -M -M -M -M -M 0.85 -M -M -M -M -M -M -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 0.5 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 0.5 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M 0.8 -M -M 0.8 -M 0.8 -M -M -M 0.8 -M -M -M -M -M
-M 0.6 0.5 0.8 0.9 -M -M -M -M -M -M -M -M -M -M -M -M 0.8 -M -M 0.8 -M -M 0.8 -M -M -M 0.8 -M -M -M 0.7
-M -M -M -M -M -M -M 0.85 0.85 0.8 -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 -M -M -M 0.85 -M 0.85 0.85 0.8 -M 0.8 -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 0.5 0.8 -M 0.85 -M 0.85 0.85 0.8 -M 0.8 -M -M 0.8 -M -M 0.8 -M -M 0.8 -M 0.8 -M -M -M 0.8 -M -M -M -M -M
-M 0.6 0.5 0.8 0.9 0.85 -M 0.85 0.85 0.8 -M 0.8 -M -M 0.8 -M -M 0.8 -M -M 0.8 -M 0.8 -M -M -M 0.8 0.8 -M -M -M 0.7
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Table 3 Optimal xp}q of the tourism company problem. i;j Links
ðA; BÞ ðA; FÞ ðA; GÞ ðA; JÞ ðA; LÞ ðB; AÞ ðB; CÞ ðB; EÞ ðC; BÞ ðC; DÞ ðD; CÞ ðD; EÞ ðE; BÞ ðE; DÞ ðE; FÞ ðF; AÞ ðF; EÞ ðF; GÞ ðG; AÞ ðG; FÞ ðG; HÞ ðH; GÞ ðH; IÞ ðH; JÞ ðI; HÞ ðJ; AÞ ðJ; HÞ ðJ; KÞ ðK; JÞ ðK; LÞ ðL; AÞ ðL; KÞ
Tourism Groups A}C
A}E
A}F
A}I
A}K
C}E
C}F
C}I
C}K
2 0 – – – – 2 0 – – 0 – – 0 – – 0 – – – – – – – – – – – – – – –
3 0 – – – – – 3 – – – – – – – – 0 – – – – – – – – – – – – – – –
– 0 2 – – – – – – – – – – – – – – – – 2 – – – – – – – – – – – –
– 2 0 3 – – – – – – – – – – – – – 2 – – 2 – 5 – – – 3 – – – – –
– 3 0 0 0 – – – – – – – – – – – – 3 – – 3 – – 3 – – – 3 – – – 0
– – – – – – – 1 1 0 – 0 – – – – – – – – – – – – – – – – – – – –
– 0 – – – 0 – 1 1 0 – 0 – – 1 – – – – – – – – – – – – – – – – –
– 0 0 0 – 0 – 3 3 0 – 0 – – 3 – – 3 – – 3 – 3 – – – 0 – – – – –
– 0 0 2 0 2 – 0 2 0 – 0 – – 0 – – 0 – – 0 – 0 – – – 0 2 – – – 0
The amount of supply (or demand) of the tourism groups are presented in Table 1. Also, cp}q are reliability for travel on link i;j ði; jÞ for tourism group p}q and these values can be estimated from historical accident data-base. The estimation of link reliability are given by Table 2. Note that cp}q is set to be M, which M is a sufficiently large positive value, to avoid from exisi;j tence of positive cycles in the network, because existence of such a cycle can make the problem unbounded. In fact, it can be interpreted that if cp}q equals to M then there exist no pathway for tourism group p}q from node i to node j. i;j If only finding the shortest paths is the goal, then Floyd–Warshall algorithm [2] can be applied to determine the shortest paths between all pairs of nodes. But, in addition to this objective, the safety of clients is important for tourism company. In fact, both of transfer cost minimization and transfer safety maximization must be considered simultaneously by the company. To do so, one can use the multi-objective programming approaches. But in most cases, there is no optimal solution that optimizes both of the objectives. A simpler way is to optimize a ratio of the objectives. Therefore, the company can convey its clients from the origins to the destinations in such a way that the ratio of the safety to the travel cost is maximized. For this aim, tourism company can solve the maximum fractional multi-commodity flow problem 1.1,1.2. We solved this problem considering the values given in this section, a ¼ 0 and b ¼ 1, by solving its corresponding linear problem. The optimal solution is presented in Table 3. Also, the optimal solution of the linear dual problem is given in Table 4. One can easily check the complementary slackness conditions of Theorem 3.1 for this problem.
5. Conclusion and future directions In this paper a fractional multi-commodity flow problem is studied and based on its linear programming representation, the dual of this problem is defined and some duality properties are derived with a new approach. Also strong complementary slackness theorem for a linear fractional problem is proved which is the first time based on the best of our knowledge. This property is an interesting and really important property, much useful to expand interior point methods to solve this class of optimization problems (see e.g., [15]). There is another issue that must be noticed. It was explained that to apply the duality theory of linear fractional programming problems, it is no necessary to assume boundedness of the primal feasible set. In fact, it is sufficient to assume solvability of the primal problem. So it will be useful to research, in future works, on the subject of finding some weak sufficient conditions which guarantee solvability of primal fractional problems.
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A. Fakhri, M. Ghatee / Applied Mathematical Modelling 38 (2014) 2151–2162 Table 4 ; hi;j ; zÞ. Optimal dual solution of the tourism company problem: ðpp}q i
pA}C A pA}C B pA}C C pA}C D pA}C E pA}C F pA}C G pA}C H pA}C I pA}C J pA}C K pA}C L pA}E A pA}E B pA}E C pA}E D pA}E E pA}E F pA}E G pA}E H pA}E I pA}E J pA}E K pA}E L pA}F A pA}F B pA}F C pA}F D pA}F E pA}F F pA}F G pA}F H pA}F I pA}F J pA}F K pA}F L
-5.7861 6.0467 5.2267 5.5718 6.1960 5.8318 10.3638 14.2674 87.0658 11.4376 10.8502 8.5185 7.5177 7.7783 5.7634 7.3827 7.8216 7.5069 9.9235 12.0660 89.2244 8.2504 6.3607 4.1455 9.5723 0.9399 2.3077 1.7948 1.8879 8.9076 8.6970 14.4702 85.7987 14.3520 13.7454 11.5211
pA}I A pA}I B pA}I C pA}I D pA}I E pA}I F pA}I G pA}I H pA}I I pA}I J pA}I K pA}I L pA}K A pA}K B pA}K C pA}K D pA}K E pA}K F pA}K G pA}K H pA}K I pA}K J pA}K K pA}K L pC}E A pC}E B pC}E C pC}E D pC}E E pC}E F pC}E G pC}E H pC}E I pC}E J pC}E K pC}E L
-13.7840 4.9014 2.6854 3.8722 7.7907 13.1070 13.2552 13.8123 219.1784 13.2088 10.5730 3.6544 7.2112 5.7159 9.7990 10.6514 9.0071 6.5342 6.6824 7.2395 93.6077 7.8430 7.7589 7.2557 4.8537 8.5846 9.4046 9.1586 8.6279 10.5895 8.8447 9.0464 91.7181 5.8514 5.2827 5.9099
pC}F A pC}F B pC}F C pC}F D pC}F E pC}F F pC}F G pC}F H pC}F I pC}F J pC}F K pC}F L pC}I A pC}I B pC}I C pC}I D pC}I E pC}I F pC}I G pC}I H pC}I I pC}I J pC}I K pC}I L pC}K A pC}K B pC}K C pC}K D pC}K E pC}K F pC}K G pC}K H pC}K I pC}K J pC}K K pC}K L
-8.4269
hA;B
0
8.1325
hA;F
0
8.9525
hA;G
0
8.6982
hA;J
0
8.1758
hA;L
0
7.7970
hB;A
0
9.0016
hB;C
0
9.5517
hB;E
0.0209
91.1229
hC;B
0
7.9666
hC;D
0
7.3365
hD;C
0
7.9622
hD;E
0
13.0671
hE;B
0
12.7844
hE;D
0
13.6043
hE;F
0
13.3486
hF;A
0
12.8276
hF;E
0
12.4488
hF;G
0.0624
12.5971
hG;A
0
13.1541
hG;F
0
219.8366
hG;H
0.0465
12.5245
hH;G
16.7365
hH;I
0 233.6499
16.8598
hH;J
0
18.1615
hI;H
0
18.4221
hJ;A
0
17.6021
hJ;H
0
0.4778
hJ;K
0
18.2101
hK;J
0
34.9170
hK;L
0
51.6413
hL;A
0
67.0079
hL;K
0
z
0.3929
1.4095 18.7368 18.8209 18.7625
Furthermore the complementary slackness optimality conditions are obtained for fractional multi-commodity flow problem which is the base for a lot of optimization algorithms which can be followed in future researches. Also it is possible to investigate on extending some algorithms based on the lagrangian relaxation on fractional multi-commodity flow problems applying the results of this paper. Acknowledgement The first author is grateful to Dr. M. Soleimani-damaneh (School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran) for his valuable comments. References [1] S.S. Chadha, V. Chadha, Linear fractional programming and duality, Cent. Eur. J. Oper. Res. 15 (2007) 119–125. [2] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory and Applications, Prentice-Hall, Englewood cliffs, 1993. [3] H. Salehi Fathabadi, S. Khodayifar, M.A. Raayatpanah, Minimum flow problem on network flows with time-varying bounds, Appl. Math. Model. 36 (2012) 4414–4421. [4] F. Jimenez, J.L. Verdegay, Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach, Eur. J. Oper. Res. 117 (1999) 485–510. [5] M. Ghatee, S.M. Hashemi, Some concepts of the fuzzy multicommodity flow problem and their application in fuzzy network design, Math. Comput. Model. 49 (2009) 1030–1043. [6] S. Schaible, Fractional programming, in: R. Horst, P.M. Pardalos (Eds.), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, Boston, London, 1995, pp. 495–608.
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[7] S. Schaible, J. Shi, ‘‘Recent developments in fractional programming: single ratio and max-min case’’, in: W. Takahashi, T. Tanaka (Eds.), Proceedings of the 3rd International Conference in Nonlinear Analysis and Convex Analysis, 2004, pp. 2–11. [8] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, 2006. [9] A. Fakhri, ‘‘Duality and nonlinear fractional programming’’, Unpublished M.Sc. Thesis (in Persian), School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, 2011. [10] C. Xu, X.M. Xu, H.F. Wang, The fractional minimal cost flow problem on network, Optim. Lett. 5 (2011) 307–317. [11] H.D. Sherali, On a fractional minimal cost flow problem on networks, Optim. Lett. 6 (2012) 1945–1949. [12] A. Charnes, W.W. Cooper, Programming with linear fractional functionals, Naval Res. Logistics Quart. 9 (1962) 181–196. [13] S. Schaible, Duality in fractional programming: a unified approach, Oper. Res. 24 (1976) 452–461. [14] M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and Network Flows, John Wiley & Sons, 1990. [15] J.W. Stephen, Primal-dual interior-point methods, SIAM, 1997.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Fractional multi-commodity flow problem: Duality and optimality conditions q Ashkan Fakhri, Mehdi Ghatee ⇑ Department of Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran Intelligent Transportation Systems Research Institute, Amirkabir University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 24 June 2012 Received in revised form 31 August 2013 Accepted 8 October 2013 Available online 29 October 2013 Keywords: Combinatorial optimization Fractional programming Duality Strong complementary slackness Multi-commodity flow problem
a b s t r a c t This paper deals with multi-commodity flow problem with fractional objective function. The optimality conditions and the duality concepts of this problem are given. For this aim, the fractional linear programming formulation of this problem is considered and the weak duality, the strong direct duality and the weak complementary slackness theorems are proved applying the traditional duality theory of linear programming problems which is different from same results in Chadha and Chadha (2007) [1]. In addition, a strong (strict) complementary slackness theorem is derived which is firstly presented based on the best of our knowledge. These theorems are transformed in order to find the new reduced costs for fractional multi-commodity flow problem. These parameters can be used to construct some algorithms for considered multi-commodity flow problem in a direct manner. Throughout the paper, the boundedness of the primal feasible set is reduced to a weaker assumption about solvability of primal problem which is another contribution of this paper. Finally, a real world application of the fractional multi-commodity flow problem is presented. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction In many application contexts, several physical commodities, vehicles, or messages, each governed by their own network flow constraints, share the same network [2,3]. Multi-commodity flow problem is achieved in a lot of network design and transportation problems [4]. In this problem some commodities from special origins should be transmitted to some other destinations with a minimum total cost [2]. Solving this problem under integer restriction is NP-hard [2]. When multiple objective functions or uncertain ones are given, the solution process of multi-commodity problem is harder than traditional methods [5]. When only two objective functions are considered into account e.g., cost minimization and reliability maximization, one can optimize the ratio of these objective functions. The provided fractional programming problem can be solved applying some famous approaches. Besides, fractional programs are happened in a lot of practical problems [6,7] which can be pursued in other network analysis. For fractional multi-commodity problems to satisfy the demand for each commodity at each node without violating the constraints imposed by the supply–demand and capacity, one can consider the following
q
This paper was partially supported by Intelligent Transportation Systems Research Institute, Amirkabir University of Technology, Tehran, Iran.
⇑ Corresponding author at: Department of Computer Science, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran. Tel.: +98 21 64542510; fax: +98 21 66497930. E-mail addresses: [email protected] (A. Fakhri), [email protected] (M. Ghatee). URL: http://www.aut.ac.ir/ghatee (M. Ghatee). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.10.032
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maximum fractional multi-commodity flow model in which G ¼ ðN; AÞ is a network with N and A as the sets of n nodes and m links:
PK P k¼1
maximize f ðxÞ ¼ PK P k¼1
k k ði;jÞ2A c i;j xi;j k k ði;jÞ2A di;j xi;j
þa þb
ð1:1Þ
;
8P P k > xk j:ðj;iÞ2A xkj;i ¼ bi ; 8i 2 N; 8k ¼ 1; . . . ; K; > < Pj:ði;jÞ2A i;j K k s:t: 8ði; jÞ 2 A; k¼1 xi;j 6 ui;j ; > > : k xi;j P 0; 8ði; jÞ 2 A; 8k ¼ 1; . . . ; K:
ð1:2Þ
Here, xki;j is a nonnegative variable regarding to the amount of flow of kth commodity which streams through link ði; jÞ; ui;j is k the capacity of link ði; jÞ and K is the number of commodities. Moreover, for commodity k; cki;j and di;j are the unit-reliability k and the unit-cost of flow through link ði; jÞ, respectively, and bi is the supply or demand of node i, defined by positive or negative numbers. In this model a and b are two positive constants regarding two lower levels for the corresponding objective functions. Applying vector and matrix notations, this model can be rewritten as the following maximum fractional linear programming model: t
a maximize f ðxÞ ¼ dc t xþ xþb
ð1:3Þ
s: t: x 2 S1 ¼ fx : Ax 6 b; x P 0g: Here A is a ð2:n:K þ mÞ by ðm:KÞ matrix as follows:
k¼1
k¼2
k ¼ K
ði; jÞ 2 A ði; jÞ 2 A 2
0
N
6 N 6 6 6 0 6 6 0 6 A¼6 6 .. 6 . 6 6 0 6 6 4 0 I
ði; jÞ 2 A
; 0
0 N
0
0 0
N .. .
0 .. .
.. .
0 .. .
0
N
0
N
I
I
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
8i 2 N 8i 2 N 8i 2 N 8i 2 N .. .
;
ð1:4Þ
8i 2 N 8i 2 N 8ði; jÞ 2 A
in which N is the n by m incidence matrix of the network G and I is the m by m identity matrix. Also b is a column vector with 2:n:K þ m components as follows: 1 3 b 6 17 6 b 7 6 2 7 6 b 7 7 6 6 27 6 b 7 7 b¼6 6 .. 7 6 . 7 7 6 6 bK 7 7 6 6 K7 4 b 5
2
k
k¼1 k¼2 k¼2 ; .. .
ð1:5Þ
k¼K k¼K ði; jÞ 2 A
u k
k¼1
k
k t
where b ¼ ½b1 ; b2 ; . . . ; bn is supply–demand vector with respect to commodity k, for k ¼ 1; . . . ; K, and x; c and d are column k vectors with m:K components including the values of xki;j ; cki;j and di;j , respectively. Also, it is assumed that S1 is nonempty and t d x þ b > 0 for all x in S1 . In addition, we assume that the optimal solution of fractional multi-commodity flow problem (1.1) and (1.2) is bounded. Note that the objective function f ðxÞ is a pseudo-linear (both pseudo-convex and pseudo-concave) function and any local optimal solution of problem (1.3) is also a global optimal solution [8]. For this problem, Chadha and Chadha [1] proposed a linear dual problem and investigated weak duality, strong direct duality and weak complementary slackness theorems (see [9], also). In [10] a network simplex algorithm and also a similar dual problem is proposed for this problem with a single commodity namely a fractional minimal cost flow problem. In continuation, Sherali [11] showed that the resulted theorems
A. Fakhri, M. Ghatee / Applied Mathematical Modelling 38 (2014) 2151–2162
2153
and duality introduced in [10] can be directly found by linear programming transformations. This paper follows the Sherali’s approach for obtaining same results for multi-commodity flow problem. It is worthwhile to mention that in [1,10] there is no reference to the strong complementary slackness theorem. In fact, a direct proof for this theorem seems to be more complicated than the ones for the other duality relations. For treating this difficulty, we show in Section 2, via some straightforward results, that the dual problem proposed by Chadha and Chadha [1] can be derived by duality theory of linear programming. After presenting the new proofs for duality relations, we can prove indirectly the strong complementary slackness theorem applying the traditional duality principles in linear programming when well-known Charnes and Cooper [12] variable transformation is done on problem (1.3). Since the results of this paper are derived only by solvability assumption on the primal problem (1.3), the proposed theorems are more applicable than previous works based on boundedness of S1 . In fact, our assumption is weaker because if S1 is bounded then the problem (1.3) is reduced to a problem in which a continuous function should be maximized over a compact set, which certainly is a solvable problem (see, e.g., [8]). Based on these new results on fractional linear programming, the dual problem of fractional multi-commodity flow problem is presented in Section 3 and a new reduced cost is defined to check optimality of solutions. Finally the complementary slackness conditions are proved and a real world application is introduced for fractional multi-commodity flow problem. Section 4 ends this paper with some brief remarks. 2. Duality properties of fractional linear programming problem By the Charnes–Cooper variable transformation [12], t
1
t ¼ ðd x þ bÞ ;
t
1
w ¼ ðd x þ bÞ x;
x 2 S1 ;
one can rewrite the problem (1.3) as the following problem:
maximize hðw; tÞ ¼ ct w þ at; s:t:
n o t ðw; tÞ 2 S2 ¼ ðw; tÞ : Aw bt 6 0; d w þ bt ¼ 1; w P 0; t > 0 :
ð2:1Þ
After replacing the last constraint with t P 0, the following linear programming problem can be achieved:
maximize hðw; tÞ ¼ ct w þ at; s: t:
n o t ðw; tÞ 2 S3 ¼ ðw; tÞ : Aw bt 6 0; d w þ bt ¼ 1; w P 0; t P 0 :
ð2:2Þ
The following two lemmas indicate some relationships among problems (1.3), (2.1) and (2.2). Lemma 2.1 (see [13]). The problem (1.3) has optimal solution if and only if the problem (2.1) has one. In fact if x is an optimal 1 1 t t solution of problem (1.3) then ðw; tÞ ¼ ððd x þ bÞ x; ðd x þ bÞ Þ is an optimal solution of problem (2.1), and if ðw; tÞ is an optimal solution of problem (2.1) then x ¼ ðw=tÞ is an optimal solution of problem (1.3). Obviously, in the both cases optimal values are equal.
Lemma 2.2. If ðw ; t Þ solves problem (2.1) then ðw ; t Þ solves problem (2.2) too. Conversely, if ðw ; t Þ solves problem (2.2) and t > 0 then ðw ; t Þ solves problem (2.1) too. ^ ^tÞ 2 S3 such Proof. Assume that ðw ; t Þ solves problem (2.1). In contradiction to desired result, suppose that there exists ðw; that
^ þ a^t > ct w þ at ct w
^ ^tÞ > hðw ; t ÞÞ: ði:e: hðw;
ð2:3Þ
For each k 2 ð0; 1Þ, we have
^ ^tÞ þ ð1 kÞðw ; t Þ 2 S2 ðwk ; t k Þ :¼ kðw;
ð2:4Þ
^ þ a^t ct w at Þ þ ct w þ at > ct w þ at ¼ hðw ; t Þ; hðwk ; tk Þ ¼ ct wk þ at k ¼ kðct w
ð2:5Þ
and
where the last inequality of (2.5) is valid because of (2.3). But the combination of (2.4) and (2.5) contradicts optimality of ðw ; t Þ for problem (2.1). Hence ðw ; t Þ solves problem (2.2) too. Conversely, if ðw ; t Þ solves problem (2.2) and t > 0 then ðw ; t Þ solves problem (2.1) because objective functions of the problems (2.1) and (2.2) are similar, S2 # S3 and ðw ; t Þ 2 S2 . h The following corollary is directly deduced from Lemmas 2.1 and 2.2:
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t
t
1
Corollary 2.1. If x solves problem (1.3) then ðw ; t Þ ¼ ððd x þ bÞ x ; ðd x þ bÞ Þ solves problem (2.2) and f ðx Þ ¼ hðw ; t Þ. Conversely, if ðw ; t Þ solves problem (2.2) and t > 0 then x ¼ ðw =t Þ solves problem (1.3) and f ðx Þ ¼ hðw ; t Þ. Now consider the linear dual of the problem (2.2) (see, e.g., [14]):
minimize gðy; zÞ ¼ z; s:t:
n o t ðy; zÞ 2 S4 ¼ ðy; zÞ : At y þ dz P c; b y þ bz P a; y P 0 :
ð2:6Þ
t
After replacing the inequality b y þ bz P a with equality, the following problem can be derived from problem (2.6):
minimize gðy; zÞ ¼ z s:t:
n o t ðy; zÞ 2 S5 ¼ ðy; zÞ : At y þ dz P c; b y þ bz ¼ a; y P 0 :
ð2:7Þ
which is exactly similar to the dual form proposed by Chadha and Chadha [1]. Lemma 2.3 reveals some relationships among optimal values of the preceding problems. Lemma 2.3. Assume that problem (1.3) is solvable. Then (i) Problem (2.6) is solvable and its optimal value is equal to optimal value of problem (2.2) (and hence equal to optimal value of problem (1.3), regarding Corollary 2.1. ; zÞ of problem (2.6) is a feasible solution of problem (2.7). (ii) Any optimal solution ðy ^; ^zÞ is optimal for problem (2.6) if and only if it is optimal for problem (2.7). (iii) ðy
^ ^tÞ that solves problem Proof. ðiÞ Let ^ x be an optimal solution of problem (1.3). Then regarding Corollary 2.1, there exists ðw; (2.2) and ^t > 0. Hence according to the strong duality theorem of linear programming [14], problem (2.6) has an optimal solution with the same optimal value of problem (2.2). ; zÞ is optimal for problem (2.6). According to Corollary 2.1, problem (2.2) has an optimal solution ðiiÞ Assume that ðy ðw ; t Þ so that t > 0. Therefore, applying the weak complementary slackness theorem of linear programming [14], we have t þ bz ¼ a. This completes the proof of part ðiiÞ. b y ^; ^zÞ is optimal for problem (2.6). Then as a result of the part ðiiÞ; ðy ^; ^zÞ is feasible for problem (2.7) and ðiiiÞ Assume that ðy hence optimal for it (because objective functions of problems (2.6) and (2.7) are similar and S5 # S4 ). Conversely, assume that ^; ^zÞ is optimal for problem (2.7). Obviously ðy ^; ^zÞ is feasible for problem (2.6). In cotradiction to desired result, suppose that ðy ^; ^zÞ is not optimal for problem (2.6). Then there exists an optimal solution ðy ; zÞ of problem (2.6) such that z < ^z (existence ðy ; zÞ is feasible for problem (2.7) too (see of optimal solution for problem (2.6) is guaranteed by the part (i)). But in this case, ðy ^; ^zÞ for problem (2.7). This contradiction completes the proof. h part ðiiÞ), which contradicts optimality of ðy In the sequel, we prove duality theorems. Note that Theorems 2.1, 2.2 and 2.3 have been proved by Chadha and Chadha [1] with another approach. However, in what follows, these theorems are proved applying the traditional duality theorems for linear programming problems. In addition, in Theorems 2.4 and 2.5, strong complementary slackness property is stated which is the first time based on the best of our knowledge. Also in Theorem 3.1 the optimality conditions for fractional multicommodity flow problem are presented. Theorem 2.1 (Weak Duality Theorem [1]). For each x in S1 and for each ðy; zÞ in S5 , we have f ðxÞ 6 gðy; zÞ. t
1
t
1
Proof. Let x belongs to S1 . Then ðw; tÞ ¼ ððd x þ bÞ x; ðd x þ bÞ Þ is in S3 and f ðxÞ ¼ hðw; tÞ. Regarding weak duality theorem of linear programming (see, e.g., [14]):
hðw; tÞ 6 z 8ðy; zÞ 2 S4 : But S5 S4 and hence
f ðxÞ ¼ hðw; tÞ 6 z ¼ gðy; zÞ 8ðy; zÞ 2 S5 : This completes the proof. h Next theorem is strong direct duality theorem which is a direct corollary of Lemma 2.3. Theorem 2.2 (Strong Direct Duality Theorem [1]). If problem (1.3) if solvable then problem (2.7) is solvable and their optimal values are equal.
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Proof. See Lemma 2.3. h The following example illustrates that if problem (2.7) is solvable then problem (1.3) is not necessarily solvable. Example 2.1. Consider the following problem:
maximize
x1 x1 þx2
s:t:
ð2:8Þ
x1 P 1; x2 P 0: Then, ðy ; z Þt ¼ ð0; 0Þt solves the dual of (2.8), but (2.8) has not optimal solution.
Theorem 2.3 (Weak Complementary Slackness Theorem [1]). Let u and v be slack and surplus column vectors associated with ^ Þ and ðy ^; ^z; v ^ Þ be feasible x; u problems (1.3) and (2.7), respectively, and assume that problem (1.3) is solvable. Furthermore, let ð^ ^ Þ solves problem (1.3) and ðy ^; ^z; v ^ Þ solves problem (2.7) if and solutions of problems (1.3) and (2.7), respectively. Then, ð^ x; u only if
v^ t ^x þ u^t y^ ¼ 0: ^; u ^ Þ solves problem (1.3) and ðy ^; ^z; v ^ Þ solves problem (2.7). According to Corollary 2.1, Proof. First assume ðx 1 1 t t ^; ^z; v ^ Þ solves ðw ; t Þ ¼ ððd ^ x þ bÞ ^ x; ðd ^ x þ bÞ Þ solves problem (2.2). On the other hand, due to the part (iii) of Lemma 2.3, ðy problem (2.6). Hence applying the weak complementary slackness theorem of linear programming [14] one can state: t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ w ¼ 0: ðAw bt Þ y
ð2:9Þ
Dividing the Eq. (2.9) by t > 0, we have t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ ^x ¼ 0: ðA^x bÞ y
ð2:10Þ
Therefore
^t y ^ ¼ 0 and u
v^ t ^x ¼ 0
ð2:11Þ
and hence
^t y ^ þ v^ t ^x ¼ 0: u
ð2:12Þ
^; u ^ and v ^ , we have Conversely, assume that u y þ v x ¼ 0. Because of nonnegativity of ^ x; y ^t ^
^t y ^ ¼ 0 and u
^t^
v^ t ^x ¼ 0
ð2:13Þ
and hence t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ ^x ¼ 0: ðA^x bÞ y
ð2:14Þ t
Multiplying the above equations by positive t ¼ 1=ðd ^ x þ bÞ and defining w :¼ t ^ x, we have
ðw ; t Þ 2 S3 ;
^; ^zÞ 2 S4 ; ðy
ð2:15Þ t
t ^ ¼ 0 and ðAt y ^ þ d^z cÞ w ¼ 0: ðAw bt Þ y
ð2:16Þ
Now with respect to the weak complementary slackness theorem of linear programming [14], ðw ; t Þ solves problem (2.2) ^; ^zÞ solves problem (2.6). Therefore ^ ^; ^zÞ solves problem (2.7) (see and ðy x solves problem (1.3) (see Corollary 2.1) and and ðy part (iii) of Lemma 2.3). This completes the proof. h ^ Þ solves problem (1.3) and ðy ^; ^z; v ^ Þ solves problem (2.7) if and Remark 2.1. By the same assumptions of Theorem 2.3, ð^ x; u only if
v^ a :^xa ¼ 0; ^ b :y ^b ¼ 0; u where a and b are two arbitrary indices for the corresponding vectors. The following theorem is one of the main results of this paper. Note that we have not supposed the feasible set S1 to be bounded, see [9].
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Theorem 2.4 (Strong Complementary Slackness Theorem). Assume that problem (1.3) is solvable, and let u and v be slack and ^ Þ and ðy ^; ^z; v ^Þ surplus column vectors associated with problems (1.3) and (2.7), respectively. Then there exist optimal solutions ð^ x; u for problems (1.3) and (2.7), respectively, such that
^þy ^ > 0 and u
v^ þ ^x > 0:
ð2:17Þ
Proof. Regarding Corollary 2.1, problem (2.2) is solvable. Now with respect to the strong complementary slackness theorem ^; ^zÞ of problems (2.2) and (2.6), respectively, such that of linear programming [14], there exist optimal solutions ðw ; t Þ and ðy
^ þ ðbt Aw Þ > 0; y
ð2:18Þ
^ þ d^z cÞ > 0; w þ ðAt y
ð2:19Þ
t ^ þ b^z aÞ > 0: t þ ðb y
ð2:20Þ t
^; ^zÞ is an optimal solution of problem (2.7). Hence b y ^ þ b^z a ¼ 0, and relation According to part (iii) of Lemma 2.3, ðy (2.20) can be rewritten as follows:
t > 0:
ð2:21Þ t
^ :¼ A y ^ þ d^z c, we have ^ Dividing the relations (2.18) and (2.19) by t and defining ^ x :¼ w =t and v x 2 S1 and
^=t Þ þ ðb A^xÞ > 0 and ^x þ ðv^ =t Þ > 0; ðy ^ :¼ ðb A^ in which ^ x is an optimal solution of problem (1.3) (see Corollary 2.1. Now defining u xÞ, we have
^=t Þ þ u ^ > 0 and ^x þ ðv^ =t Þ > 0; ðy
ð2:22Þ
or equivalently
^þu ^ > 0 and ^x þ v^ > 0; y
ð2:23Þ
^; u ^ and v ^ . Therefore we have found soluwhere (2.23) is a direct corollary of (2.22), positivity of t and nonnegativity of ^ x; y ^ Þ and ðy ^; ^z; v ^ Þ as optimal solutions for problems (1.3) and (2.7), respectively, such that ^ ^; y ^ and v ^ satisfy (2.17). tions ð^ x; u x; u This completes the proof. h
Remark 2.2. Solvability of problem (1.3) is crucial, because as we illustrated in Example 2.1, if problem (2.7) is solvable then problem (1.3) is not necessarily solvable. Note that replacing the unrestricted variable z by two nonnegative variables zþ and z , problem (2.7) can be rewritten as:
minimize zþ z s: t:
ð2:24Þ þ
At y þ dz dz P c;
t
b y þ bzþ bz ¼ a;
y; zþ ; z P 0:
Quite similar to Theorem 2.4, we can state the following result: Theorem 2.5. Assume that problem (1.3) is solvable, and let u and v be slack and surplus column vectors associated with problems ^ Þ and ðy ^; z^þ ; z^ ; v ^ Þ for problems (1.3) and (2.24), respectively, (1.3) and (2.24), respectively. Then, there exist optimal solutions ð^ x; u such that
^þy ^ > 0; u
v^ þ ^x > 0
and z^þ ; z^ > 0:
ð2:25Þ
3. Optimality conditions for fractional multi-commodity flow problem In this section, we consider the fractional multi-commodity flow problem (1.1) subjected to (1.2) and try to find some optimality conditions for this problem. Doing so permits us to assess whether or not we have found an optimal solution to the problem. It is also permits us to interpret various algorithms as particular methods for solving the optimality conditions, and in several instances even suggests novel algorithmic approaches for solving the problem we were studying [2]. Since the fractional multi-commodity flow problem is a fractional linear program, based on the duality theory of linear programming, we can present the dual of the fractional multi-commodity flow problem (1.1) and (1.2). Because of different bundle constraints for every link ði; jÞ, mass balance constraints for each node-commodity combination and an auxiliary constraint, the dual of this problem has three types of variables: a price hi;j on each link ði; jÞ, node potential pki for each combination of commodity k and node i and also an auxiliary variable z related to the auxiliary constraint. Applying these dual p variables, it is possible to define the reduced cost ck; i;j of link ði; jÞ with respect to the commodity k as follows:
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p k k k ck; i;j :¼ ðc i;j di;j z hi;j Þ pi þ pj ; k
ð3:1Þ
where
z¼
! K X X X 1 k aþ bi pki þ ui;j hi;j : b ði;jÞ2A k¼1 i2N
ð3:2Þ
The following theorem states that a primal feasible solution xki;j and a dual feasible solution ðhi;j ; z; pki Þ are optimal to the respective problems if they satisfy a new set of complementary slackness conditions. Theorem 3.1 (Fractional Multi-commodity Flow Complementary Slackness Conditions). Assume that the fractional multicommodity flow problem (1.1) subjected to (1.2) is solvable. The commodity flows xki;j are optimal for this problem if and only if they are feasible and for some nonnegative link prices hi;j , (unrestricted is sign) node potentials pki and scalar z, the following complementary slackness conditions are satisfied:
! K X k xi;j ui;j hi;j ¼ 0; for all links ði; jÞ 2 A;
ð3:3Þ
k¼1
p xki;j ck; i;j ¼ 0; for all links ði; jÞ 2 A and all k ¼ 1; . . . ; K;
ð3:4Þ
p ck; i;j 6 0; for all links ði; jÞ 2 A and all k ¼ 1; . . . ; K;
ð3:5Þ
in which,
p ck; i;j
and z are defined by Eqs. (3.1) and (3.2), respectively.
Proof. To prove this theorem, we consider matrix form of fractional multi-commodity flow problem represented in (1.3). Noting to Eqs. (1.4) and (1.5), the nonnegative dual variables piþ;k and p;k can be defined for a couple of mass balance coni straints for each node-commodity combination ði; kÞ and also the nonnegative variables hi;j with respect to the capacity constraint on link ði; jÞ. According to problem (2.7), the dual problem of fractional multi-commodity flow problem can be stated as follows:
minimize gðh; z; pÞ ¼ z s: t: k
þ;k k ðpþ;k p;k p;k 8k ¼ 1; . . . ; K; 8ði; jÞ 2 A; i i Þ ðpj j Þ þ hi;j þ di;j z P c i;j ; ! K X X k X k ;k X bi pþ;k þ bi pi ui;j hi;j þ bz ¼ a; i k¼1
i2N
i2N
ð3:6Þ
ði;jÞ2A
;k pþ;k P 0; 8i 2 N; i ; pi
hi;j P 0;
8ði; jÞ 2 A:
Applying a variable transformation in which the unrestricted variable pki is used instead of pþ;k p;k i i , the dual of fractional multi-commodity flow problem can be stated as follows:
minimize gðh; z; pÞ ¼ z s: t:
pki pkj þ hi;j þ dki;j z P cki;j ; 8k ¼ 1; . . . ; K; 8ði; jÞ 2 A;
ð3:7Þ
K X X X k bi pki ui;j hi;j þ bz ¼ a; k¼1 i2N
hi;j P 0;
ði;jÞ2A
8ði; jÞ 2 A:
Now, note that the desired result is a straightforward corollary of Theorems 2.2 and 2.3.
h
Theorem 3.2. Let the commodity flows xki;j be optimal in the fractional multi-commodity flow problem (1.1) and (1.2), hi;j be the optimal link prices and pki be optimal node potentials with respect to commodity k. Then for each commodity k, the flow variables xki;j solve the following (uncapacitated) fractional maximum cost flow problem:
P ck xk þa max Pði;jÞ2A i;jk i;jk
d x þb ði;jÞ2A i;j i;j
s:t:
ð3:8Þ N xk ¼ b k ; xki;j P 0; for all ði; jÞ 2 A;
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are defined as the following: in which, cki;j and a
cki;j ¼ cki;j hi;j and
a ¼ a þ
K X X
k0 b i2N i
k0 ¼1 k0 –k
0
pki þ
X
u h ði;jÞ2A i;j i;j
Proof. The structure of proof is similar to [2]. Since xki;j are optimal flows, hi;j are optimal link prices and pki are optimal node potentials for the fractional multi-commodity flow problem, these variables together with some z, defined by Eq. (3.2), satisfy the complementary slackness conditions (3.3)–(3.5). But (3.4) and (3.5) are the optimality conditions for the fractional . h maximum cost flow problem for commodity k with link costs cki;j and numerator constant a Attempting to find a sequential method to solve the fractional multi-commodity flow problem is postponed to our future works. Finally, in what follows, we present a real world example of problem (1.1). This example is only one of the several applications of the presented fractional multi-commodity flow problem. 4. Tourism company application A tourism company wishes to select appropriate and reasonable pathways to transfer its clients (tourists) from some origins to some destinations in northern cities of Iran near to Caspian see depicted in Fig. 1. There are different kinds of nodes: A is an origin of tourists, C is both origin and destination, E; F; I and K are destinations and the others are middle cities. Consider the network of Fig. 2, whose nodes are origin and destination cities of Fig. 1 and the links are corresponding to the real paths between each couple of nodes. In this example, the following terminologies and notations are used. A group of tourists who are conveyed between a certain origin p and a certain destination q, are considered as commodity p}q, and the set of such tourism groups (commodities) is denoted with K:
K ¼ fA}C; A}E; A}F; A}I; A}K; C}E; C}F; C}I; C}Kg: p}q
For each group of tourists p}q; b p}q bp
¼
p}q bq
¼b
is the number of travelers from p to q; hence for each p}q 2 K we set
p}q
and p}q
bi
cp}q i;j p}q di;j
¼ 0;
and
p}q di;j
8i – p; q: are unit-reliability and unit-cost values to travel the tourism group p}q through link ði; jÞ. It is assumed that
¼ di;j (see the link labels illustrated in Fig. 2) and ui;j ¼ 8 for each link ði; jÞ.
Fig. 1. Northern cities of Iran.
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Fig. 2. The imaginary network with respect to the topological network of Fig. 1.
Table 1 The number of travelers between different origins and destinations. Tourism groups p}q b
p}q
A}C
A}E
A}F
A}I
A}K
C}E
C}F
C}I
C}K
2
3
2
5
3
1
1
3
2
Table 2 Degrees of reliability cp}q for link ði; jÞ and tourism group p}q. i;j Links
ðA; BÞ ðA; FÞ ðA; GÞ ðA; JÞ ðA; LÞ ðB; AÞ ðB; CÞ ðB; EÞ ðC; BÞ ðC; DÞ ðD; CÞ ðD; EÞ ðE; BÞ ðE; DÞ ðE; FÞ ðF; AÞ ðF; EÞ ðF; GÞ ðG; AÞ ðG; FÞ ðG; HÞ ðH; GÞ ðH; IÞ ðH; JÞ ðI; HÞ ðJ; AÞ ðJ; HÞ ðJ; KÞ ðK; JÞ ðK; LÞ ðL; AÞ ðL; KÞ
Tourism Groups A}C
A}E
A}F
A}I
A}K
C}E
C}F
C}I
C}K
0.85 0.6 -M -M -M -M 0.85 0.85 -M -M 0.8 -M -M 0.8 -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
0.85 0.6 -M -M -M -M -M 0.85 -M -M -M -M -M -M -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 0.5 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 0.5 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M 0.8 -M -M 0.8 -M 0.8 -M -M -M 0.8 -M -M -M -M -M
-M 0.6 0.5 0.8 0.9 -M -M -M -M -M -M -M -M -M -M -M -M 0.8 -M -M 0.8 -M -M 0.8 -M -M -M 0.8 -M -M -M 0.7
-M -M -M -M -M -M -M 0.85 0.85 0.8 -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 -M -M -M 0.85 -M 0.85 0.85 0.8 -M 0.8 -M -M 0.8 -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M -M
-M 0.6 0.5 0.8 -M 0.85 -M 0.85 0.85 0.8 -M 0.8 -M -M 0.8 -M -M 0.8 -M -M 0.8 -M 0.8 -M -M -M 0.8 -M -M -M -M -M
-M 0.6 0.5 0.8 0.9 0.85 -M 0.85 0.85 0.8 -M 0.8 -M -M 0.8 -M -M 0.8 -M -M 0.8 -M 0.8 -M -M -M 0.8 0.8 -M -M -M 0.7
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Table 3 Optimal xp}q of the tourism company problem. i;j Links
ðA; BÞ ðA; FÞ ðA; GÞ ðA; JÞ ðA; LÞ ðB; AÞ ðB; CÞ ðB; EÞ ðC; BÞ ðC; DÞ ðD; CÞ ðD; EÞ ðE; BÞ ðE; DÞ ðE; FÞ ðF; AÞ ðF; EÞ ðF; GÞ ðG; AÞ ðG; FÞ ðG; HÞ ðH; GÞ ðH; IÞ ðH; JÞ ðI; HÞ ðJ; AÞ ðJ; HÞ ðJ; KÞ ðK; JÞ ðK; LÞ ðL; AÞ ðL; KÞ
Tourism Groups A}C
A}E
A}F
A}I
A}K
C}E
C}F
C}I
C}K
2 0 – – – – 2 0 – – 0 – – 0 – – 0 – – – – – – – – – – – – – – –
3 0 – – – – – 3 – – – – – – – – 0 – – – – – – – – – – – – – – –
– 0 2 – – – – – – – – – – – – – – – – 2 – – – – – – – – – – – –
– 2 0 3 – – – – – – – – – – – – – 2 – – 2 – 5 – – – 3 – – – – –
– 3 0 0 0 – – – – – – – – – – – – 3 – – 3 – – 3 – – – 3 – – – 0
– – – – – – – 1 1 0 – 0 – – – – – – – – – – – – – – – – – – – –
– 0 – – – 0 – 1 1 0 – 0 – – 1 – – – – – – – – – – – – – – – – –
– 0 0 0 – 0 – 3 3 0 – 0 – – 3 – – 3 – – 3 – 3 – – – 0 – – – – –
– 0 0 2 0 2 – 0 2 0 – 0 – – 0 – – 0 – – 0 – 0 – – – 0 2 – – – 0
The amount of supply (or demand) of the tourism groups are presented in Table 1. Also, cp}q are reliability for travel on link i;j ði; jÞ for tourism group p}q and these values can be estimated from historical accident data-base. The estimation of link reliability are given by Table 2. Note that cp}q is set to be M, which M is a sufficiently large positive value, to avoid from exisi;j tence of positive cycles in the network, because existence of such a cycle can make the problem unbounded. In fact, it can be interpreted that if cp}q equals to M then there exist no pathway for tourism group p}q from node i to node j. i;j If only finding the shortest paths is the goal, then Floyd–Warshall algorithm [2] can be applied to determine the shortest paths between all pairs of nodes. But, in addition to this objective, the safety of clients is important for tourism company. In fact, both of transfer cost minimization and transfer safety maximization must be considered simultaneously by the company. To do so, one can use the multi-objective programming approaches. But in most cases, there is no optimal solution that optimizes both of the objectives. A simpler way is to optimize a ratio of the objectives. Therefore, the company can convey its clients from the origins to the destinations in such a way that the ratio of the safety to the travel cost is maximized. For this aim, tourism company can solve the maximum fractional multi-commodity flow problem 1.1,1.2. We solved this problem considering the values given in this section, a ¼ 0 and b ¼ 1, by solving its corresponding linear problem. The optimal solution is presented in Table 3. Also, the optimal solution of the linear dual problem is given in Table 4. One can easily check the complementary slackness conditions of Theorem 3.1 for this problem.
5. Conclusion and future directions In this paper a fractional multi-commodity flow problem is studied and based on its linear programming representation, the dual of this problem is defined and some duality properties are derived with a new approach. Also strong complementary slackness theorem for a linear fractional problem is proved which is the first time based on the best of our knowledge. This property is an interesting and really important property, much useful to expand interior point methods to solve this class of optimization problems (see e.g., [15]). There is another issue that must be noticed. It was explained that to apply the duality theory of linear fractional programming problems, it is no necessary to assume boundedness of the primal feasible set. In fact, it is sufficient to assume solvability of the primal problem. So it will be useful to research, in future works, on the subject of finding some weak sufficient conditions which guarantee solvability of primal fractional problems.
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A. Fakhri, M. Ghatee / Applied Mathematical Modelling 38 (2014) 2151–2162 Table 4 ; hi;j ; zÞ. Optimal dual solution of the tourism company problem: ðpp}q i
pA}C A pA}C B pA}C C pA}C D pA}C E pA}C F pA}C G pA}C H pA}C I pA}C J pA}C K pA}C L pA}E A pA}E B pA}E C pA}E D pA}E E pA}E F pA}E G pA}E H pA}E I pA}E J pA}E K pA}E L pA}F A pA}F B pA}F C pA}F D pA}F E pA}F F pA}F G pA}F H pA}F I pA}F J pA}F K pA}F L
-5.7861 6.0467 5.2267 5.5718 6.1960 5.8318 10.3638 14.2674 87.0658 11.4376 10.8502 8.5185 7.5177 7.7783 5.7634 7.3827 7.8216 7.5069 9.9235 12.0660 89.2244 8.2504 6.3607 4.1455 9.5723 0.9399 2.3077 1.7948 1.8879 8.9076 8.6970 14.4702 85.7987 14.3520 13.7454 11.5211
pA}I A pA}I B pA}I C pA}I D pA}I E pA}I F pA}I G pA}I H pA}I I pA}I J pA}I K pA}I L pA}K A pA}K B pA}K C pA}K D pA}K E pA}K F pA}K G pA}K H pA}K I pA}K J pA}K K pA}K L pC}E A pC}E B pC}E C pC}E D pC}E E pC}E F pC}E G pC}E H pC}E I pC}E J pC}E K pC}E L
-13.7840 4.9014 2.6854 3.8722 7.7907 13.1070 13.2552 13.8123 219.1784 13.2088 10.5730 3.6544 7.2112 5.7159 9.7990 10.6514 9.0071 6.5342 6.6824 7.2395 93.6077 7.8430 7.7589 7.2557 4.8537 8.5846 9.4046 9.1586 8.6279 10.5895 8.8447 9.0464 91.7181 5.8514 5.2827 5.9099
pC}F A pC}F B pC}F C pC}F D pC}F E pC}F F pC}F G pC}F H pC}F I pC}F J pC}F K pC}F L pC}I A pC}I B pC}I C pC}I D pC}I E pC}I F pC}I G pC}I H pC}I I pC}I J pC}I K pC}I L pC}K A pC}K B pC}K C pC}K D pC}K E pC}K F pC}K G pC}K H pC}K I pC}K J pC}K K pC}K L
-8.4269
hA;B
0
8.1325
hA;F
0
8.9525
hA;G
0
8.6982
hA;J
0
8.1758
hA;L
0
7.7970
hB;A
0
9.0016
hB;C
0
9.5517
hB;E
0.0209
91.1229
hC;B
0
7.9666
hC;D
0
7.3365
hD;C
0
7.9622
hD;E
0
13.0671
hE;B
0
12.7844
hE;D
0
13.6043
hE;F
0
13.3486
hF;A
0
12.8276
hF;E
0
12.4488
hF;G
0.0624
12.5971
hG;A
0
13.1541
hG;F
0
219.8366
hG;H
0.0465
12.5245
hH;G
16.7365
hH;I
0 233.6499
16.8598
hH;J
0
18.1615
hI;H
0
18.4221
hJ;A
0
17.6021
hJ;H
0
0.4778
hJ;K
0
18.2101
hK;J
0
34.9170
hK;L
0
51.6413
hL;A
0
67.0079
hL;K
0
z
0.3929
1.4095 18.7368 18.8209 18.7625
Furthermore the complementary slackness optimality conditions are obtained for fractional multi-commodity flow problem which is the base for a lot of optimization algorithms which can be followed in future researches. Also it is possible to investigate on extending some algorithms based on the lagrangian relaxation on fractional multi-commodity flow problems applying the results of this paper. Acknowledgement The first author is grateful to Dr. M. Soleimani-damaneh (School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran) for his valuable comments. References [1] S.S. Chadha, V. Chadha, Linear fractional programming and duality, Cent. Eur. J. Oper. Res. 15 (2007) 119–125. [2] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory and Applications, Prentice-Hall, Englewood cliffs, 1993. [3] H. Salehi Fathabadi, S. Khodayifar, M.A. Raayatpanah, Minimum flow problem on network flows with time-varying bounds, Appl. Math. Model. 36 (2012) 4414–4421. [4] F. Jimenez, J.L. Verdegay, Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach, Eur. J. Oper. Res. 117 (1999) 485–510. [5] M. Ghatee, S.M. Hashemi, Some concepts of the fuzzy multicommodity flow problem and their application in fuzzy network design, Math. Comput. Model. 49 (2009) 1030–1043. [6] S. Schaible, Fractional programming, in: R. Horst, P.M. Pardalos (Eds.), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, Boston, London, 1995, pp. 495–608.
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