Combinatorial algorithms for the minimum interval cost flow problem

Applied Mathematics and Computation 175 (2006) 1200–1216 www.elsevier.com/locate/amc Combinatorial algorithms for the minimum interval cost flow problem S. Mehdi Hashemi, Mehdi Ghatee *, Ebrahim Nasrabadi Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran Abstract The aim of minimum the interval cost flow problem (MICFP) is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy demands at certain nodes from available supplies at other nodes where there exists some vague in vector cost of problem. Interval cost is a common event in uncertainty environments, where statistical data are applied. Moreover they almost play an essential role in fuzzy programming, specially in the case of using their cuts. In this paper, a complete order on intervals is defined and efficient combinatorial algorithms for MICFP are proposed. Digital simulation results show the performance of the proposed algorithms compared with real scenarios.  2005 Elsevier Inc. All rights reserved. Keywords: Minimum interval cost flow problem; Complete ordering; Combinatorial algorithms * Corresponding author. E-mail addresses: hashemi@aut.ac.ir (S.M. Hashemi), mahdighatee@yahoo.com (M. Ghatee), nasrabadi@aut.ac.ir (E. Nasrabadi). 0096-3003/$ - see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.08.044 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 1201 1. Introduction The minimum cost flow problem (MCFP) is an important problem in combinatorial optimization and has a lot of applications in practice. As MCFP represents a general form of the network flows, the results from studying it, can be applied to many other network problems such as transportation, multi-commodity flow problem, assignment, shortest path, maximum flow, scheduling and linear programs with consecutive 1Õs in columns [1,2]. There are two essential ideas for solving the MCFP; network simplex algorithms and combinatorial methods based on network flows. Combinatorial algorithms rely heavily upon optimality conditions: negative cycle optimality conditions, reduced cost optimality conditions and complementary slackness optimality conditions. Some of these algorithms maintain primal feasible solutions and strive toward optimality, and others maintain primal infeasible solutions that satisfy the optimality conditions and strive toward feasibility. These are pseudo-polynomial time algorithms. Furthermore by utilizing scaling approach, several polynomial time algorithms may be developed, see e.g. [1,2]. On the other hand, costs, capacities or supplies–demands of the network are generally vague or uncertain in many actual cases. Cai et al. [4] proposed a dynamic scheme for the MCFP against these variations; however, interval computations, fuzzy set theory and probability methods appear more suited to conquer such vague aspects. According to these views, Shit and Lee [21] proposed a fuzzy version of MCFP using linear programming. But they could not create an efficient algorithm, as a result of non-convexity and NP-hard inherent characteristic of multi-level programming. Also, we can implement a fuzzy or interval programming as multi-objective optimization [3,19]. In this scope, Noda et al. [15,16] developed two network simplex methods for biobjective MCFP. They found all efficient extreme solutions by the non-polynomial time algorithm. Moreover, Osman et al. [17] solved this problem with integrity assumption. In this paper, interval computations are used, which is common for estimating quantities, where the prediction of their dependent factors are difficult, for instance in traffic conditions, accidents, traffic jams or weather conditions [12,23]. Also these are the base for fuzzy programming [5,8,10], when their cuts are used in computations. Now for simplicity, interval costs in the problem are supposed, named minimum interval cost flow problem (MICFP) and interval capacities, supplies and demands are left to future works. Although Das et al. [5] has proposed an algorithm for the multi-objective transportation problem with interval costs applying linear programming and finding Pareto optimal solutions, there exists a no efficient algorithm on MICFP. The proposed idea in this paper is similar to Perny and SpanjaardÕs approach [18]. They noticed that comparison of solutions in combinatorial problems is often based on an additive cost function including a complete order on solutions. 1202 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 Thus, they used a preference binary relation on the solution space, and developed some algorithms for preferred spanning trees and preferred paths. In ordering comments on intervals, Moore [14] was the first author who extended ‘‘<’’ on real line and ‘‘’’ on the sets to create two transitive order relations over intervals. Ishibuchi and Tanaka [9] as an advance over Moore, suggested two order relations 6LR and 6MW on them. However, these orderings were not complete. Kundu [11] defined a fuzzy preference relationship between two interval and used it to find the optimal decision. More differently, Sengupta and Pal [20] suggested two indexes for comparing two intervals numerically. But almost all of them are not complete and also lexicographic. Following that a complete order on intervals by employing non-algebraic numbers is created and some combinatorial algorithms for MICFP with common ideas of combinatorics are introduced. Validation of digital simulation results is performed using real scenarios, which is similar to comparisons of Montemanni et al. [12]. The rest of the paper is organized as follows: Some basic definitions and results on interval numbers are given in the next section. In Section 3 several important theorems on MICFP are obtained. Then some combinatorial algorithms for MICFP is created in Section 4. Section 5 consists of numerical simulations. Section 6 ends this paper with conclusions and future directions. 2. Preliminaries An interval number A is the set of all real numbers x, such that aL 6 x 6 aR, where aL and aR are the left and right limits of the interval A. Interval number A is usually denoted by A = [aL, aR]. Interval A is alternatively represented as R L and a ¼ aR a are the center and the width of interA = ha, ai, where a ¼ aL þa 2 2 val A, respectively. It is trivial that each real number x is a degenerate interval, which can be denoted by [x, x] or h x,0i. We review some classic arithmetic on intervals, see e.g. Moore [13] for details. Definition 2.1. For each interval A = [a, b] and B = [c, d] addition and multiplication respectively, are defined as follows: ½a; b þ ½c; d ¼ ½a þ c; b þ d; ½a; b.½c; d ¼ ½minfac; ad; bc; bdg; maxfac; ad; bc; bdg. Moreover, we need operator cancellation as defined by Hansen [7, p. 10] as follows: ½a; b  ½c; d ¼ ½a  c; b  d; where [c, d]  [a, b]. S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 1203 An ordering on intervals based on weighting scheme may be proposed as follows: For simplicity in ordering notation, up to rest of the paper second forms (hcenter, widthi representation) for intervals will used. Definition 2.2. A relation  defined on T is said to be • • • • Reflexive iff e  e for every e in T. Transitive iff for every e and f in T, if e  f and f  g then e  g. Anti-symmetric iff for every e and f in T, if e  f and f  e then e = f. Complete iff for every e and f in T, e  f or f  e. Definition 2.3. A partial order is a reflexive, anti-symmetric and transitive binary relation. Also a complete order is a reflexive, anti-symmetric, transitive and complete binary relation. Definition 2.4. For two arbitrary intervals A = ha, ai and B = hb, bi and each real positive numbers ‘‘k’’ and ‘‘l’’, less than or equal relation 6k,l may be defined as follows ha; ai6k;l hb; bi iff k.a þ l.a 6 k.b þ l.b, where kl is the ratio of importance of the center to the spread in the decision making process. Clearly risk may increase if k  l. This property permit us to model the risk averse, risk neutral, and risk seeking decision maker, easily; For risk averse investor, the weight of the center is set bigger than the weight of the width. In contrast, for risk seeking modeling the weight of the center is chosen smaller than other. Naturally for modeling the risk neutral state, k and l, are selected nearly. Proposition 2.5. Order relation 6k,l on intervals is a reflexive and transitive relation. For making a complete order, let us limit the choice of k and l. Employing non-algebraic numbers are a suitable choice, as will be mentioned soon. Nonalgebraic numbers is very important in number theory and polynomial Rings and are defined as follows. Definition 2.6. A complex number z is said to be algebraic if and only if, it is a root of a non-zero polynomial equation by integer coefficients, else it is said to be non-algebraic or transcendental. Proposition 2.7. The set of algebraic numbers are computable. So the set of nonalgebraic numbers are incompatible. 1204 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 Example. Real number ‘‘p = 3.1415. . .’’, which is defined as the ratio of a circleÕs circumference to its diameter and the constant ‘‘e = 2.7182. . .’’, which is the base of the natural logarithm, are non-algebraic. Among the hundreds of references, the reader may address Filaseta [6] for details. Proposition 2.8. Let p be a non-algebraic real positive number and k ¼ q1 pn1 and l ¼ q2 pn2 , where q1 ; q2 2 Qþ  f0g be non-zero rational numbers and n1 6¼ n2 2 f0; 1; 2; . . .g be natural numbers. Then, 6k,l on {ha, aija, a are rational numbers} is a complete order. Proof. Let A; B; C be in fha; aija; a are rational numbersg and A6k;l B and B6k;l C, immediately, A6k;l C, then 6k;l is transitive. Also A6k;l A, so it is reflexive. Now let A ¼ ða; aÞ6k;l ðb; bÞ ¼ B and B6k;l A. Then ka þ la ¼ kb þ lb; or q1 pn1 ða  bÞ þ q2 pn2 ða  bÞ ¼ 0; but (a  b) and (a  b) are rational. So by producing two sides of the equation in a sufficiently large number, an equation with integer coefficients can be obtained. So a¼b and a ¼ b; then 6k,l is anti-symmetric. As 6 is complete on real numbers, 6k,l is complete on {ha, aija, a are rational numbers} too. h Remark. Working on {ha, aija, a are rational numbers} is appropriate in practice, because only numbers by finite floating point may be registered in computations. Moreover, the double precision can be used for implementing non-algebraic numbers. 3. Minimum interval cost flow problem (MICFP) Let G = (N, A) be a directed network that N and A are sets of nodes and edges, respectively. Moreover we get a cost ci,j and a capacity of upper bound ui,j and lower bound li,j for every edge (i, j) in A. We can suppose li,j to be zero, otherwise it transforms to this form [1]. Then we can write minimum cost flow problem (MCFP) as follows: S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 min X 1205 ci;j xi;j ; ði;jÞ2A s.t. X xi;j  fj:ði;jÞ2Ag X xj;i ¼ bi ; 8i 2 N ; fj:ðj;iÞ2Ag 0 6 xi;j 6 ui;j . In the above formulation if bi < 0 we say node i 2 N is demander and if bi > 0 node i 2 N is supplier. Otherwise i is a transient node. We can suppose P i2A bi ¼ 0. Also, xi,j is the amount of flow on edge (i, j). The objective is to minimize the total cost of shipment. Minimum interval cost flow problem (MICFP) is a similar problem, except for ci,j to be supposed in {ha, aija, a are positive rational numbers}. We show this problem as follows: X hcci;j ; cwi;j ixi;j min ði;jÞ2A X s.t. fj:ði;jÞ2Ag xi;j  X xj;i ¼ bi ; 8i 2 N ; fj:ðj;iÞ2Ag 0 6 xi;j 6 ui;j . If in the above objective function we use hxi,j, 0i instead of xi,j, we obtain X X min hcci;j ; cwi;j i.hxi;j ; 0i ¼ min hcci;j xi;j ; cwi;j xi;j i. ði;jÞ2A ði;jÞ2A So this program is a bi-objective program. We want to extend some combinatorial algorithms by new ordering approach for this problem. These assumptions are naturally in the MCFP and so we suppose them for MICFP • All data (center and width of cost, supply/demand, and capacity) are integral. • The network is directed. P • The supplies/demands at the nodes satisfy the condition i2N bi ¼ 0; moreover, the minimum interval cost flow problem has a feasible solution. • All arc interval costs are non-negative, i.e. cci;j  cwi;j P 0 for each edge (i, j). Definition 3.1. X is the optimal solution of the minimum interval cost flow problem, if and only if its cost is minimum among all costs of feasible flows in the network by 6k,l ordering. Definition 3.2. It is easy to define the residual network respect to a given flow X as follows (see e.g. Ahuja et al. [1]). 1206 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 Replace each arc (i, j) in the original network by two arcs (i, j) and (j, i). The cost of (i, j) is ci,j and its residual capacity ri,j = ui,j  xi,j, and the arc (j, i) has a cost ci,j and a residual capacity rj,i = xi,j. The residual network consists of only the arcs with a positive residual capacity. We use the notation G(X) to represent the residual network; A(G(X)) and N(G(X)) denote the set of arcs and nodes of G(X) respectively. We assume for any pair of nodes i and j, graph G does not contain both arc (i, j) and arc (j, i). Then the residual network will contain no parallel arcs. But this assumption does not impose any loss of generality. For details see [1, p. 45]. Theorem 3.3. Suppose X is a feasible solution of MICFP, the following is equivalent: • X is the optimal solution of MICFP. • Negative cycle optimality conditions (NCOC): Residual network G(X) contains no negative interval cost (directed) cycle by 6k,l ordering. • Reduced interval cost optimality conditions (RICOC): There exists some set of node potentials fhpci ; pwi igi2N that satisfy the following statements: 8ði; jÞ 2 AðGðX ÞÞcpi;j ¼ hcci;j ; cwi;j i  hpci ; pwi i þ hpcj ; pwj iPk;l 0. • Complementary slackness optimality conditions (CSOC). There exists some set of potentials p, that reduced costs and flow values satisfy the following conditions: – if cpi;j >k;l 0; then xi,j = 0, – if cpi;j <k;l 0; then xi,j = ui,j. – if 0 < xi,j < ui,j, then cpi;j ¼ 0. Proof. If in proofs of Ahuja et al. [1, pp. 307–310], substitute 6k,l ordering instead of 6, the results would be obtained easily. We can write the dual of MICFP as follows: Dual MICFPðDMICFPÞ X X max bi hpci ; pwi i  ai;j ui;j i2N s.t. ði;jÞ2A hpci ; pwi i  hpcj ; pwj i  ai;j 6k;l hcci;j ; cwi;j i; pci 2 R; pwi P 0 & ai;j P 0.  We then have Theorem 3.4 (Weak duality theorem). If Z(x) be the objective value of MICFP with respect to a feasible solution x of the primal, and W(p, a) is the objective value of DMICFP with respect to a feasible solution {pi, ai, j} of the dual, then 1207 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 W ðp; aÞ6k;l ZðxÞ. Proof. It is obvious if ha, ai 6 k,l hb, bi and x P 0 then ha, ai.x 6 k,l hb, bi.x. Now by using matrix formation for the primal and the dual and by producing two sides of the dual constraint in x the result is revealed. h Property 3.5. Let cpi;j ¼ ci;j  pi þ pj where all these parameters are rational intervals i.e. in {ha, aija, a are rational numbers}. From the dual constraint ai;j Pk;l  cpi;j . On the other hand, ai,j P k,l h0, 0i. So ai;j Pk;l maxf0; cpi;j g. But the coefficient of ai,j in the objective function is ui,j and ui,j P 0. So for minimization, we must have ai;j ¼ maxf0; cpi;j g. Thus, the objective function of Dual MICFP may be rewritten as follows: X X W ðpÞ ¼ max bi hpci ; pwi i  maxf0; cpi;j gui;j . i2N ði;jÞ2A Theorem 3.6 (Strong duality theorem). If MICFP and DMICFP are feasible and x* be the optimal solution of MICFP, then DMICFP has an optimal solution p* that satisfies Zðx Þ ¼ W ðp Þ. Proof. Let x* be an optimal solution of MICFP. As mentioned earlier, there is a vector p of node potentials which satisfies CSOC. It is trivial that cpi;j xi;j ¼ maxf0; cpi;j gui;j . On the other hand, we have X cx  cp x ¼ ðpi  pj Þxi;j ¼ ði;jÞ2A ¼ X X fijði;jÞ2Ag X pi xi;j  j pi ðout flowðiÞ  in flowðiÞÞ ¼ fjjði;jÞ2Ag X X pj xi;j i pi bi . i i By Property 3.5 the result is obtained. X h Theorem 3.7. If the set of feasible solution of the MICFP is bounded from below, then the network has a spanning tree respect to the optimal solution of the MICFP. 1208 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 Proof. Similar to simplex network theorem in Ahuja et al. [1, 408]. h 4. Extending some efficient algorithm for MICFP In this section this fact is identified that we can extend network flow algorithms to interval cost types directly. First, a label-correcting algorithm for finding shortest path is proposed. It is the base in our algorithm in MICFP. Up to rest of paper, we suppose parameters in {ha, aija, a are rational numbers} for abbreviation. Definition 4.1. A path between r and s is called the shortest interval path if and only if it has the minimum cost among all paths between r and s by 6k,l ordering. Theorem 4.2 (Interval shortest path optimality condition). For every node j in N let d(j) denote the interval cost of some directed path from the source node to node j, then d(j) represents the interval shortest path if and only if they satisfy the following properties for each edge (i, j): dðjÞ6k;l dðiÞ þ ci;j . Proof. Similar to the shortest path optimality condition in Ahuja et al. [1]. h According to this theorem, a label correcting algorithm that satisfies the above condition may be created. Let 0 s 0 be the source. Then the following algorithm finds the minimum distance from 0 s 0 to each node of the network. Algorithm 4.3 (Modified label-correcting for interval cost) Input. A, C, r, s, k, l Output. d, Pred d(s) = h0, 0i, pred(s) = 0 d(j) = h1, 0i for all j 2 N{s} List = {s} While List 5 ; • Remove an element i from List • For each arc (i, j) 2 A – if d(j) > k,l d(i) + ci,j * d(j) = d(i) + ci,j * pred(j) = i * if j is not in List insert it. S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 1209 By the above algorithm for the shortest path, several efficient algorithms for MICFP can be proposed. Before we prefer to establish a theorem in complexity. The following axiom for continuance of the discussion is needed. Axiom 4.4. We agree to this upper bound definition for the set of intervals fhcci;j ; cwi;j i : ði; jÞ 2 Ag sup hcci;j ; cwi;j i6k;l ðkC c þ lC w Þ=k; ði;jÞ2A where C c ¼ maxði;jÞ2A fcci;j g and C w ¼ maxði;jÞ2A fcwi;j g. Although the above statement is trivial but with 6k,l ordering it cannot be proved, because k and l do not create members in {ha, aija, a are rational numbers}. Theorem 4.5. The complexity of Algorithm 4.3 is at most O(nmdCe) where C = (kCc + l Cw)/k. Proof. Observe that for every j 2 N, we have ndCe 6 k,ld(j) 6 k,lndCe (by the above axiom). So Algorithm 4.3 updates any label d(j) at most 2nC times, because each update of d(j) decreases it by at least 1 unit. On the other hand, whenever the algorithm updates d( j) it adds node j to the listP in a later iteration and scans its arc list. So the algorithm is repeated at most j2N ð2nCÞjAðjÞj ¼ OðnmCÞ times. h Remark 4.6. It is easy to create some more efficient algorithm by polynomial complexity for this problem. For example, implementing this algorithm by a heap structure would obtain an algorithm by O(n log(n)) complexity. For details see Ahuja et al. [1]. 4.1. Successive shortest path algorithm for MICFP In this subsection, a combinatorial method for the MICFP is described. We need some definitions as follows. Definition 4.7. Flow X is pseudo-flow iff it satisfies capacity and nonnegativity constrains on all edges. Definition 4.8. For any pseudo-flow X, defines the imbalance of node i as X X eðiÞ ¼ bðiÞ þ xj;i  xi;j . fj:ðj;iÞ2Ag fj:ði;jÞ2Ag 1210 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 If e(i) > 0, we refer to e(i) as the excess of node i, and if e(i) < 0, we refer e(i) as deficit of node i. The successive shortest path algorithm maintains optimality of the solution at every step and strives to attain feasibility. It starts by a pseudo-flow X and at each step selects a node s with an excess supply and a node t with deficit and sends flow from s to t along the shortest path in a residual network. The algorithm terminates when the current solution satisfies all the mass balance constrains. Theorem 4.9. Let X be flow (pseudo-flow) that satisfies RICOC with respect to some fhpci ; pwi igi2N , and vector d represents interval of the shortest distance between some node s and all other nodes in the residual network G(X) with cpi;j as the length of an arc (i,j), then the following properties are valid: • Flow (pseudo-flow) x also satisfies RICOC with respect to the p 0 = p  d. 0 • The reduced costs cpi;j are zero for all arcs (i, j) in an interval shortest path from s to every node. Proof. Let X satisfy the RICOC with respect to fhpci ; pwi igi2N , then cpi;j Pk;l 0 for each arc (i, j) in G(X). On the other hand, by the interval shortest path optimality condition for each arc (i, j) in G(X), we have dðjÞ 6k;l dðiÞ þ cpi;j the results would be obtained. h For an algorithmic scheme, we can start by X = 0. Clearly, the reduced cost of each edge is positive, so this pseudo-flow satisfies optimality conditions. By the above theorem, we ensure optimality in each step. A successive shortest path algorithm for interval cost may be extended, respect to this terminology, as follows, where A, C, U be an n-by-n matrix and b is an n vector. Algorithm 4.10 (MICFP by successive) Input. A, C, U, b, k, l Output. value, X p = 0, e = b, D = find(e < 0), E = find(e > 0), R = U,RC = C, test = 0; While not empty (E) and test 6 size(D) • s = E(1), t = D(1), AG = R > 0; • [Path d] = find path by algorithm 1 by these inputs (AG, RC, s, t, k, l) • If d(t) 6 k,l h 1,0i – p=pd – Delta = min(e(s), e(t), R(i, j):(i, j) in Path) S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 • • • • • • 1211 – If Delta > 0 * For each (i, j) in the path Set R(i, j) = R(i, j)Delta, R(j, i) = R(j, i) + Delta * * Set e(s) = e(s)  Delta, e(t) = e(t) + Delta * If e(s) = 0 remove s from E and test = 0 * If e(t) = 0 remove t from D * for each (i, j) in AG update RC(i,j) = C(i, j)  p(i,:) + p(j,:) – Else * Remove s from the head of D and insert s in its tail * test = test + 1 if test > size(D) print(The problem is infeasible) value = h0, 0i else for each edge (i, j) in A X(i, j) = R(j, i) value = value + hX(i, j),0i*Ci, j This algorithm finds an optimal solution, if it exists otherwise it reports infeasibility. 4.2. Negative cycle canceling for the MICFP In this subsection, we generalize negative cycle canceling for interval cost. This algorithm maintains a feasible solution and attempts to improve its objective function value. It iteratively finds directed cycles with a negative interval cost in the residual network and augments flow on these cycles. The algorithm terminates when the residual network contains no negative cycle. Sokkalingam et al. [22] composed this idea with scaling on arc capacities in residual networks and established a polynomial time algorithm for MICFP. We combine their algorithm by our ordering and obtain an efficient algorithm for the MICFP. A starting feasible solution X for the network may be found, for example, employing maximum flow problem [1]. Algorithm 4.11 (Scaling negative cycle canceling for the MICFP) Input. A, C, U, X, k, l Output. value, Xopt Umax = max(U), d = 2mUmax, p = 0, cpi;j ¼ ci;j for each (i, j) in A • ri,j = Ui,j  Xi,j, rj,i = Xi,j • While d P 1/2 • eligible = [] 1212 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 • for each (i, j) in A, if ri,j P d – Insert (i, j) to eligible • test = 0 • While not empty (eligible) and test 6 size(eligible) + 1 – Remove [p, q] from top of eligible – found = 0, queue = [q], S = [0,. . .,0] – AðdÞ ¼ fði; jÞ 2 AG  ðq; pÞjC pi;j Pk;l dg – While not empty (queue) * Remove i from the queue and set S(i) = 1 * for j from 1 to n Æ if S(j) = 0 and ri,j P d then Insert i to the queue – if S(p) = 0 for j from 1 to n p * if S(j) = 1 set pj ¼ pj  cp;q * found = 0; – else p * L = (Li, j), Li;j ¼ maxfci;j ; 0g * [w,d] = find path by algorithm 1 by these inputs (A(d), L, q, p, k, l) * dmax = max{d(j)jS(j) = 1} * for j from 1 to n if S(j) = 1 set pj = pj + dmax  d(j) p * if ci;j 6k;l 0 Æ set test = 0, found = 1, w(p, q) = 1 else set test = test + 1, found = 0 * if found =1 * Æ H = min{ri,jjw(i, j) = 1} Æ for each (i, j) in A if w(i, j) = 1 set ri,j = ri,j  H, rj,i = rj,i + H Æ for each (i, j) if ri,j > 0 Æ set ci,jp = ci,j  pi + pj Æ if ci,jp < k,l0 and ri,j P d then insert (i, j) to eligible p * rmax ¼ maxfri;j jci;j < 0g * if d > rmax/2 set d = rmax/2; else set d = d/2 value = h0, 0i for each edge (i, j) in A • set Xopt(i,j) = R(j,i) • set value = value + hXopt(i, j), 0i*ci,j This algorithm find optimal solution if there exists. Theorem 4.12. The scaling cycle canceling algorithm solves the MICFP in Oðmðm þ n logðnÞÞ logðnU max ÞÞ. Proof. Similar to Sokkalingam et al. [22] by substituting our ordering instead of common ordering on real numbers the result would be obtained. h S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 1213 5. Numerical simulation If MICFP has an optimal finite value, each of our proposed algorithms and each similar algorithm, which utilize 6k,l ordering, can be found. Research on real time of these algorithms and their complexity are interesting subjects. Also, especially in uncertain cases, experiment on the quality of algorithm predictions from the optimal solution is very important. In this section, we want to compare the results of the proposed algorithms by random environment. Algorithms was implemented by MATLAB 6.5. In the first experiment, a random non-sparse network by integer coefficients and random positive interval cost with n = 10 nodes and m = 10.(10  1)/2 = 45 edges were established and the proposed algorithms on this network were executed. For understanding the rate of accuracy of these algorithms in prediction of optimal paths in a random environment, some scenarios as similar as proposed by Montemanni [12], except to its theoretical aspects, were employed. We say network G(r) is a scenario if and only if its edge costs be selected in corresponding interval costs. These new costs denoted by C r ¼ ðcri;j Þ. Each scenario has a realization aspect of arc costs of network with respect to the random environment. According to these networks, two performance indexes for the proposed algorithm may be defined as follows: • Let V be the optimal interval cost in the MICFP. Then we determine the rate of scenarios that their optimal cost is in interval V. As much as this index is closed to unit, Algorithm 4.10 and Algorithm 4.11 predict the cost of shipment more accurately. In this case, the algorithm has found good paths in an uncertainty environment. This index is denoted by I(1). • Let X be the optimal flow in the MICFP and Yr be the optimal flow of scenario r. The maximum difference between edge flow in X and in Yr can be computed. The maximization on multiplication of this number in normalized per unit cost of relating edge, among all scenarios is defined as the second performance index, i.e. ( ) cri;j r . Ið2Þ ¼ max max jX i;j  Y i;j j. r ði;jÞ2A maxfcri;j jði; jÞ 2 Ag Opposite to the first index, when this index is closed to zero, algorithms act more successfully in routing. We applied the successive interval minimum cost flow algorithm on the above mentioned network, and repeated classic algorithm for the minimum cost flow on E = 10,000 scenarios. For every experiment we varied ‘‘k’’ and ‘‘l’’ and obtained the following results (Table 1). We supposed 100 to the maximum capacity. So for a fixed network and a lot of scenarios, the proposed algorithms do not have a large dependency on k and l, and they can predict the path in 1214 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 Table 1 Results of the MICFP in the random network by 10 nodes and its performance indexes Weight of center (k) Weight of width (l) I(1) I(2) 1 1 1 1 1 1 1 1 1 1 1 4p 3p 2p p p/2 p/4 p/8 p/16 p/32 p/64 p/128 0.93260000000000 0.94170000000000 0.94050000000000 0.94020000000000 0.93550000000000 0.93680000000000 0.93620000000000 0.94310000000000 0.93720000000000 0.93610000000000 0.93910000000000 11 11 11 11 11 11 11 11 11 11 11 Table 2 Results of the MICFP in the random network by 20 nodes and its performance indexes Weight of center (k) Weight of width (l) I(1) I(2) 1 1 1 1 1 1 1 1 1 1 1 4p 3p 2p p p/2 p/4 p/8 p/16 p/32 p/64 p/128 0.57410000000000 0.58370000000000 0.57200000000000 0.59490000000000 0.58900000000000 0.58430000000000 0.58850000000000 0.58570000000000 0.57580000000000 0.57810000000000 0.58050000000000 7.600000000000000 8 8 7.61904761904762 8 8 8 8 8 8 8 uncertainty networks by almost 10% flow on the non-optimal edge. We test the same experiment on the second network by n = 20 nodes and m = 20(20  1)/ 2 = 190 edges by with 100 units as maximum capacity. We apply E = 10,000 scenarios and obtain the following results (Table 2). This experiment shows that related to improving in one index, for example I(2), the other index is not necessarily improved, because these indexes are mutually exclusive. 6. Conclusion Interval programming is an important area in modeling uncertainty in the field of decision making. In this paper a complete ordering on intervals was proposed, which used non-algebraic numbers instead of coefficients relating to the center and the width of intervals. Based on this approach some optimal- S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 1215 ity conditions for the minimum interval cost flow problem (MICFP) was obtained. According to these extensions, several combinatorial algorithms for finding the minimum interval cost of shipment are proposed. Numerical simulation showed acceptable performance in random problems. These algorithms may be extended to the fuzzy version by some variation. Moreover computation results revealed that the selection process for k and l in ‘‘6 k,l’’ ordering in the case of a large number of scenarios has no essential role. Studying interval capacities, supplies and demands and comparison between real time of algorithms will be dealth with the future work. Acknowledgement We would like to thank to Dr. M. R. Bank Tavakoli for his interesting discussion. References [1] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows, Prentice-Hall, Englewood Cliffs, 1993. [2] M.S. Bazarra, J.J. Jarvis, H.D. Sherali, Linear Programming and Networks Flow, John Wiley & Sons, New York, 1990. [3] J.J. Buckley, T. Feuring, Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming, Fuzzy Sets and Systems 109 (2000) 35–53. [4] X. Cai, D. Sha, C.K. Wong, Time-varying minimum cost flow problems, European Journal of Operational Research 131 (2001) 352–374. [5] S.K. Das, A. Goswami, S.S. Alam, Multiobjective transportation problem with fuzzy interval cost, source and destination parameters, European Journal of Operational Research 117 (1999) 100–112. [6] M. Filaseta, Algebric number theory, Instructor notes, University of South Carolina, 1999. [7] E. Hansen, Global Optimization Using Interval Analysis, Marcel Dekker Inc., USA, 1992. [8] M. Inuiguchi, J. Rammic, T. Tanino, M. Vlach, Satisficing solution and duality in interval and fuzzy linear programming, Fuzzy Sets and Systems 135 (2003) 151–177. [9] M. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research 48 (1990) 219–225. [10] S.T. Liu, C. Kao, Solving fuzzy transportation problems based on extension principle, European Journal of Operation Research 153 (2004) 661–674. [11] S. Kundu, Min-transitivity of fuzzy leftness relationship and its application to decision making, Fuzzy Sets and Systems 86 (1997) 357–367. [12] R. Montemanni, L.M. Gambardella, A.V. Donati, A branch and bound algorithm for the robust shortest path problem with interval data, Operation Research Letters 32 (2004) 225– 232. [13] R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1969. [14] R.E. Moore, Method and Application of Interval Analysis, SIAM, Philadelphia, 1979. [15] A.S. Noda, C.G. Mortin, An algorithm for the biobjective integer minimum cost flow problem, Computers and Operation Research 28 (2001) 139–156. 1216 S.M. Hashemi et al. / Appl. Math. Comput. 175 (2006) 1200–1216 [16] A.S. Noda, C.G. Martin, Gutierrez, The biobjective undirected two-commodity minimum cost flow problem, European Journal of Operational Research 164 (2005) 89–103. [17] M.S. Osman, O.M. Saad, A.G. Hassan, Solving a spatial class of large-scale fuzzy multiobjective integer linear programming problems, Fuzzy Sets and Systems 107 (1999) 289–297. [18] P. Perny, O. Spanjaard, A preference-based approach to spanning trees and shortest paths problems, European Journal of Operational Research 162 (2005) 584–601. [19] M.S. Sakawa, Fuzzy Sets and Iterative Multiobjective Optimization, Plenum Press, NY, 1993. [20] A. Sengupta, T.K. Pal, Theory and methodology on comparing interval numbers, European Journal of Operational Research 127 (2000) 28–43. [21] S.H. Shit, E.S. Lee, Fuzzy multi level minimum cost flow problems, Fuzzy Sets and Systems 107 (1999) 159–176. [22] P.T. Sokkalingam, R.K. Ahuja, J.B. Orlin, New polynomial-time cycle-canceling algorithms for minimum cost flows, Networks 36 (2000) 53–63. [23] An online reference about interval programming. Available from: <http://interval.louisiana.edu/preprints.html>.