A note on asymptotic formulae for one-imensional network flow problems Carlos F. Daganzo an Karen R. Smilowitz (to appear in Annals of Operations Research) Abstract This note evelops asymptotic formulae for single-commoity network flow problems with ranom inputs. The transportation linear programming problem (TLP) where N points lie in a region of R is one example. It is foun that the average istance travele by an item in the TLP increases with N / ; i.e., the unit cost is unboune when N an the length of the region are increase in a fixe ratio. Further, the optimum istance oes not converge in probability to the average value. These one-imensional results are a useful stepping stone towar a network theory for two an higher imensions. Key Wors: Transportation problem; istance approximations. Introuction Asymptotic formulae exist for the traveling salesman problem, TSP, (Eilon et al., 97, Karp, 977, Daganzo, 984a), an the vehicle routing problem, VRP, (Eilon et al., 97, Daganzo, 984b, Haimovich et al., 985, Newell an Daganzo, 986 an 986a). The results apply to problems where N points are ranomly an uniformly istribute on a region of a metric plane with area A, an ensity δ = N A. In all cases the istance travele per point for the TSP, or the etour istance per point for the VRP, tens to a fixe multiple of δ as N an A are increase in a fixe ratio. This limiting result hols for problems in K-imensional regions. This note presents relate but qualitatively ifferent results for single-commoity network flow problems with ranom inputs on unirecte paths. The transportation linear programming problem, TLP, with points in a region of R is an example of this class of problems. One imensional TLP s arise in connection with the earthwork minimization problem of highway esign.. Backgroun Institute of Transportation Stuies an Department of Civil an Environmental Engineering University of California, Berkeley CA 947; aganzo@ce.berkeley.eu. Department of Inustrial Engineering an Management Sciences, Northwestern University.
Here the TLP is efine as follows. Given are N points, inter-point istances, { ij, i,j=..n; i j}, satisfying the triangle inequality, an net supplies, v i, at each point. Positive v i are interprete as supplies an negative v i as emans. The goal is to fin shipments, {v ij, i,j=..n; i j}, that minimize the total istance travele while satisfying flow balance constraints. (TLP) min z = ij v ij (a) i, j N s. t.: ( vij v ji ) vi i N (b) j N v i, j N. (c) ij Equations (b) specify flow-conservation at each point, ensuring that the net flow from a point i never excees the net supply at i. Because istances satisfy the triangle inequality, in an optimum solution origin noes only emit flow an estination noes only receive flow. A TLP is feasible only if v, as can be seen by summing (b) across i. For balance problems constraints (b) are iniviually satisfie as pure equalities; for unbalance problems, they are not. We efine an always-feasible problem, ATLP, that inclues a fictitious source, i =, with the least positive net supply to make the problem feasible, an istances, i, j = M >> sup( ij ). This net supply is v = v i, if the problem is infeasible, an v =, otherwise. The fictitious istances impose a penalty for items not shippe. Since M is large, the istances satisfy the triangle inequality, an the ATLP is a TLP. If the original TLP is infeasible then the ATLP is a balance TLP, an the outflow from the fictitious source is v. The penalty component of an optimum solution is v M, an the istance component is = z vm, with = N v v i. If the TLP i= is feasible, then = z; otherwise we take the istance component of the ATLP as its solution. In the TLP/ATLP it is assume that excess supplies are left at the origins. We consier a variant, calle epot-tlp, or DTLP, where excess supplies are carrie to the extra point, or epot. In the DTLP the epot istances o not have to be fixe or large but must be non-negative,, ), i, j, an satisfy the triangle inequality. The minimum of the DTLP objective function, z D N i= i N + ( j i i, is enote D. Due to the penalty epot istances, it shoul be clear that the following is true (a formal proof can be foun in Daganzo an Smilowitz, ). Proposition : (DTLP as an upper boun to TLP). For any TLP an any associate DTLP, Furthermore, if the TLP is balance then = D. D.
3 3. Formulae This section evelops formulae for the averages of an D over a set of solutions when conitions vary. Points are embee in the norme linear space R an ientifie by a single coorinate, x i. Distance is ij = x x. We examine balance an unbalance versions of the TLP an DTLP. We i j also consier gri problems where the net supplies change as ranom variables, but points are fixe on a - D gri, an ranom problems where points are locate ranomly. The moifiers G or R for gri or ranom, an U or B for unbalance or balance, ientify problem characteristics. 3. Balance TLP problems Cumulative supply v x p ) Cumulative supply l A B C D x p x epot v ( x Fig. a presents cumulative supply vs. position, v ( = v for a TLP(B) with ranom point locations. xi x Figure. Graphical solutions of -D problems (a) TLP(B); (b) DTLP(U) i Result. (Formula for TLP(B)). The minimum cost for TLP(B) is the absolute area between an the x-axis; i.e., + N = v x = xi ) xi+ i= ( x. () i Proof: It suffices to show that () is both an upper boun an a lower boun for. For any point, x p, such as the one in Fig. a, the net flow across x p in any feasible solution is x p ) because the aggregate supply an eman on both sies of x p must be satisfie. Thus, x p ) x is a lower boun to the optimal istance travele in any small interval, (x p, x p + where is constant. Therefore, the sum on the right
4 sie of () is a lower boun for. Conversely, a feasible solution can be constructe by consiering horizontal slices of v items an transporting the items from points where the slice intersects a rising portion of curve to ajoining points where it intersects a falling portion. In the figure, v items woul be carrie from A to B an from C to D. Thus, the summation of the slices for small v (still given by ()) is the istance of a feasible solution an an upper boun to. 3.. Gri problems If points are on a gri with lattice spacing l, then () reuces to = l N i= x i ). We examine the expecte value of for TLP(B,G) with ranom net supplies. For any homogeneous, balance problem the net supplies must have zero mean, v =, the same absolute mean v = µ, an the same variance, i v = σ. The (equal) covariances must be v i v i = σ /( N ) since this covariance ensures that the i variance of the sum of all the net supplies is zero. The general formulae for the mean, absolute mean, an variance of the sum of the first i net supplies, x i ), are then: i v ( xi ) =, xi ) = iµ, an xi ) = iσ (3) N Assume now that the v i have a joint multinormal istribution, an recall that if X is a normal ranom variable with zero-mean, X = [ ( X ) / π ] v ( i = ( ) x ) π i σ i N i. Thus, in the multinormal case, we have:. (4) Result. (Expecte optimal cost of TLP(B,G)). For the homogeneous, zero-mean TLP(B,G) with normal eman in R, ( ) N i BG = σl i =. (5a) π i N Furthermore, the limit of this expression for N is 3 π BG σln. (5b) 3
5 Equation (5b) is true because the right han sie of (5a) is a Riemann sum that becomes a efinite integral for N, an such integral reuces to (5b). The etails are as follows: ( x ) N 3 σ l x x ( x ) π x x σln π N = N N N 3 π = σln. (6) 3 In terms of the point ensity, δ = /l, the average istance per point, p = N, is: BG BG / π 3 p BG σδ N. (7) This function increases without limit with the number of points, unlike in the TSP, where the average istance per point tens to a limit. The epenence with arising from the flow balancing requirements. N is cause by the long-range interactions 3.. Extensions: ranom problems an non-normal emans Equations (5b) an (7) are general an hol if the v i are not normal, but satisfy the regularity conitions of Cramér s version of the central limit theorem for large eviations. Let F be the c..f. of for a given N an let N with a fixe interval length, L. Then, is the sum of xn/l net supplies, an the central limit theorem applies. Recall L = x. Let Φ be the c..f. of the normal approximation to. Note that = F ( ) ( z) z + [ F ( z)] z Φ( z) z + [ Φ( z)] z. The central limit theorem v x for large eviations (see corollary on p.553 of Feller, 97) guarantees that the error in v ( can be mae smaller than any ε > for all v (. Thus, the error in is boune by ε L. QED Equations (5b) an (7) also hol if point locations vary in a segment of length L as a homogeneous Poisson process with rate δ an an average of N points (N = δ L). Since is a compoun Poisson ranom v ( ( ) / π variable, the central limit theorem applies an [ ]. It suffices to show x x σ N because then the expectation of (), = x, is (6). Let i( be L L the number of points in [, x], an note the conitional ranom variable ( ( i( ) ( ) + v has zero mean an variance ( ) = i( i( i( σ. For zero mean, the unconitional variance is the expectation of N
6 the conitional variance; i.e. i( N i( N x Nx x x = σ = σ + N N N L( N ) N L L L x x σ N. QED L L 3. Unbalance TLP problems (gri an ranom point locations) An exact expression for TLP(U) is more ifficult to obtain, yet it is shown in Daganzo an Smilowitz () that p = O(N / ), implying that p is also O(N / ) for unbalance problems. BG p UG Equation (7) hols for DTLP(B) since DTLP(B) TLP(B). The expression for DTLP(U) is ifferent, but qualitatively similar. To erive it, first efine the cumulative eman for the epot v '( as shown in Fig. b; i.e., v ( = v H ( x ), where H is the Heavisie unit step function an x is the epot location. ' x Since the DTLP(U) is a balance TLP with cumulative eman, Result applies with substitute for, an we have: UG Result 3. (Deterministic DTLP). For both gri an ranom problems, D + = x. (8) For the DTLP(U), the expectation of the integran of (8) is symmetric with respect to the location of the epot. Thus, for an integration region [, L] it can be simplifie as follows: L L / = x = x = x, which from (4) is L / L / / [( / ) ( Nx / L) ] / x L = x π σ = 4 9π σ LN /, i.e., we have Result 4. (Expecte optimal cost of DTLP(U)). 3 4 4 σ ln, an p σ ln. (9) 9π 9π Note that in all cases, = O(N 3/ ) when one hols l constant as N is increase. However, = O(N / ) if one hols the total region size, L = ln, constant.
7 3.. Convergence for unbalance TLP cost approximations The form of convergence of is examine next. Unlike the TSP (9) because its coefficient of variation tens to a positive constant as N. oes not converge in probability to Result 5. (Asymptotic coefficient of variation of the optimal istance for DTLP(U)) The asymptotic coefficient of variation tens to a positive constant as N increases; i.e., / constant >. Proof: Recall is a Riemann sum of the istance in small intervals of [, L], with a secon moment, = xy = xy. By symmetry, LL = v xy L / L / LL L/ L ( + v x v y v y y x ( ( ) [ ( ) ' ( )]. For large N, curve has L / inepenent increments. Thus, the integran of the secon integral involves the prouct of inepenent quantities since the absolute net supplies being multiplie correspon to non-overlapping regions of [, L]. The expression can be rewritten using the prouct of the expectations as: L / L / L / L = xy + ( [ v' ( ] y x L / L / L / L / L / ) = xy + ( y xy. The secon inequality follows from symmetry. Note now that the first integran satisfies: v ( = v ( v ( + cov{ v (, }. Thus, L / L / ) = 4 ( y xy + co, ) xy L / L / The covariance integran is strictly positive for all x an y not equal to, because an share the net supplies from to min(x,. An expression for co v (, ) is of the form σ N ρ(x/l, y/l), an ρ is positive if both its arguments are positive. The secon moment is then: = 4 ( y xy L / L / ) + σ N L / L / ρ ( x / L, y / L) xy L / L / ) = 4 ( y xy σ L ( x', y' ) x ' y ' + N / / ρ
8 Note the secon ouble integral is a positive constant; the secon term is a positive multiple of σ N L an the first term is. The secon moment becomes = + ρ o σ NL, for some ρ o >, an the variance is = ρ o σ NL = ρ o σ l N 3. Since 4 σ 9π ln 3, the coefficient of variation is inepenent of σ, l an N, an it tens to a positive constant as state. 4. Conclusions an comparison with other TLP problems It is interesting to compare these results with those for other TLP versions (see Smilowitz an Daganzo, ). The result, p = / N = O( N ) when one hols δ constant, oes not exten to planar problems or problems in higher imensions. The epenence on N is ampe in higher imensions. In - D, the epenence is only of orer log(n). In 3-D, p is boune an the stanar eviation of the optimum istance per point eclines with N. This implies that in -D the optimum istance converges in probability to the expecte value. It is also possible to show that the asymptotic -D formulae epen on region shape, even for a constant region size. If one consiers a region consisting of two intervals of equal length, L/, separate by a greater istance, D, with the epot in the mile, then p epens on D asymptotically. To see that this is true, note that in the optimal solution the total istance travele by the epot flow increases linearly with DN /, since this flow is proportional to N /. This quantity cannot be neglecte because it is shown in Section 3 that the internal istances within each sub-zone are O(N / ) when region size is hel constant. Fortunately for practical applications, shape oes not have an asymptotic effect in -D. Thus, the -D TLP is quite similar to the TSP in that the optimum istance per point is shape-inepenent an appears to converge in probability to the asymptotic mean. Although the -D TLP istance per point is unboune when one hols ensity constant (unlike in the case of the TSP), sai istance increases with N so slowly that it may be treate as a constant for problems where N only varies by a factor of. 5. Acknowlegements Research supporte in part by the University of California Transportation Center. 6. References Daganzo, C.F. (984a) The length of tours in zones of ifferent shapes, Trans. Res. B 8B, 35-46. Daganzo, C.F. (984b) The istance travele to visit N points with a maximum of C stops per vehicle: An analytic moel an an application, Transportation Science 8 (4), 33-35.
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