- Many of my models are not pure networks (i.e. an LP with an embedded network or a network with side constraints - these are essentially the same, the difference is a matter of gradation),
- LP solvers are actually quite good in solving network models,
- Some models can be be converted to pure networks, but this can make the model less straightforward and more removed from the original problem

To make it interesting, let's consider a large, sparse network. This means not all nodes have a direct link between them. Also assume a directed graph. Directed graphs are more prevalent in practical situation, and undirected graphs are not very easy to handle. A minimum cost flow problem can be stated as the following:\[ \begin{align} \min & \sum_{(i,j)\in A} c_{i,j} x_{i,j}\\ & \sum_{(j,i)\in A} x_{j,i} + \mathit{supply}_i = \sum_{(i,j)\in A} x_{i,j} + \mathit{demand}_i & \forall i \\ & x_{i,j} \in [0,u_{i,j}] & \forall (i,j) \in A \end{align}\] Here \(x_{i,j}\) is the flow from \(i \rightarrow j\). The constants \(supply_i\) and \(demand_i\) are exogenous in- and out-flow. If we want to push a flow of one from node 1 to node 2 then this would look like: \[\mathit{supply}_i = \begin{cases} 1 & \text{if $i=1$}\\ 0 & \text{otherwise}\end{cases}\] and \[\mathit{demand}_i = \begin{cases} 1 & \text{if $i=2$}\\ 0 & \text{otherwise}\end{cases}\] The LP model is sometimes respresented as \[ \begin{align} \min & \sum_{(i,j)\in A} c_{i,j} x_{i,j}\\ & \sum_{(j,i)\in A} x_{j,i} - \sum_{(i,j)\in A} x_{i,j} = b_i & \forall i \\ & x_{i,j} \in [0,u_{i,j}] & \forall (i,j) \in A \end{align}\] where \(b_i = \mathit{demand}_i - \mathit{supply}_i\).

#### GAMS model

A complete GAMS model can look like:

set i 'nodes' /n1*n5000/;alias(i,j);** network topology (directed arcs)* approx 5% density*set
a(i,j) 'arcs
i -> j';a(i,j) = uniform(0,1)<0.05; a(i,i) = no;** data*parametersc(i,j) 'cost coefficients (over arcs)'cap(i,j) 'arc capacities'supply(i) 'exogenous inflow (sources)'/ (n1,n2) 1 / demand(i) 'exogenous outflow (sinks)'/ (n3,n4) 1 / ; c(a) = uniform(1,10); cap(a) = uniform(0,0.5); positive variable x(i,j) 'flow';variable z 'objective variable';* arc capacitiesx.up(a) = cap(a); equationsflowbal 'inflow = outflow'obj 'objective'; obj.. z =e= sum(a,c(a)*x(a));flowbal(i).. sum(a(j,i),x(a)) + supply(i) =e=sum(a(i,j),x(a)) + demand(i);* reduce printingoption
limrow=0,limcol=0;model m /all/;m.solprint=0; solve m
minimizing z using lp;display
x.l; |

We have \(n=5,000\) nodes. Arcs are modeled by a two-dimensional subset a(i,j). This set is sparse: only relatively few arcs exist. The step to generate approximately \(0.05 n^2\) takes a few seconds: we draw and compare \(n^2\) random numbers. We actually have 1,250,314 entries afterwards. This is indeed sparse:

Partial view of two dimensional set a(i,j) |

We have two source nodes (\n1,n2\) and two sink nodes \(n3,n4\). The matrices

**cap**and

**c**and the vectors

**demand**and

**supply**are also sparse, The flow balance equation is heavily dependent on the sparse set

**a**. Given a node \(i\) the first summation finds all incoming arcs, i.e. nodes \(j\) such that \(j \rightarrow i\) is an existing arc. The second summation finds all leaving arcs: nodes \(j\) such that \(i \rightarrow j\) is an existing arc. This equation is quite close to what we have in the mathematical model.

The model is not small. We have:

MODEL STATISTICS BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 5,001 BLOCKS OF VARIABLES 2 SINGLE VARIABLES 1,250,315 NON ZERO ELEMENTS 3,750,943

As the capacities are somewhat small, the solution is somewhat complex:

---- 49 VARIABLE x.L flow n3 n4 n102 n490 n635 n875 n1072 n1209 n1405 n1 0.049 0.199 0.014 n2 0.359 n490 0.199 n635 0.139 n875 0.048 n1405 0.359 n1518 0.028 n1719 0.097 0.139 0.158 n1873 0.007 n1888 0.164 n2466 0.018 n2490 0.048 n2619 0.158 n3599 0.128 n3631 0.318 n4012 0.337 + n1518 n1719 n1873 n1888 n1894 n1896 n2412 n2466 n2490 n1 0.394 0.007 0.246 n2 0.113 0.048 n102 0.049 n1894 0.028 n3169 0.028 n4524 0.018 n4558 0.114 + n2619 n3169 n3599 n3619 n3631 n4012 n4489 n4524 n4558 n1 0.072 0.018 n2 0.028 0.337 0.114 n1072 0.158 n1209 0.014 n2412 0.113 n3619 0.072 n4489 0.014 n4636 0.246 + n4636 n1896 0.246

Some timings from my Windows laptop vs NEOS:

laptop | NEOS Server | |
---|---|---|

Operating System | Windows | Linux |

GAMS Generation Time | 26.406 | 12.816 |

Cplex Iterations | 319 | 319 |

Cplex Objective | 8.3763 | 8.3763 |

Cplex Time | 9.390 | 35.117 |

Somehow NEOS is relatively slow when solving. I think timing programs on NEOS is rather difficult with all that is happening on their machines.

It is noted that a pure network like this solves quickly. If I use COIN CBC instead of Cplex, we see excellent performance:

COIN-OR Branch and Cut (CBC Library 2.9) written by J. Forrest Calling CBC main solution routine... 0 Obj 0 Primal inf 3.9999996 (4) 175 Obj 8.0839851 Primal inf 1.4230331 (22) 318 Obj 8.3762647 Optimal - objective value 8.3762647 Optimal objective 8.376264693 - 318 iterations time 4.372 Solved to optimality.

#### AMPL Models

AMPL has two ways to express network models. The first is equation based and is basically the same as the GAMS model above.

param n := 5000; set I := 1..n; set A := setof{i in I, j in I: Uniform(0,1)<0.05 and i<>j} (i,j); param c{A} = Uniform(1,10); param cap{A} = Uniform(0,0.5); param supply{I} default 0; param demand{I} default 0; let supply[1] := 1; let supply[2] := 1; let demand[3] := 1; let demand[4] := 1; var x{(i,j) in A} >= 0, <= cap[i,j]; minimize Cost: sum{(i,j) in A} c[i,j]*x[i,j]; flowbal{i in I}: supply[i] + sum{(j,i) in A} x[j,i] = demand[i] + sum{(i,j) in A} x[i,j]; solve; display {(i,j) in A: x[i,j] > 0} (x[i,j]);

The model looks quite nice,

I had some problems printing the results. In some cases funny results were printed with set elements I did not recognize.. A fragment is:

: 993 994 995 996 997 998 999 1000 1001 1002 $11 $12 $13 $14 $15 $16 $17 $18 := # $4 = 1014 # $5 = 1015 # $6 = 1016 # $7 = 1017

This is not normal output and looks like a bug when printing with options to suppress empty rows. In the end I worked around the problem and just printed the nonzero values using an expression.

AMPL also has special facilities to express networks. Let's see how that works with this example.

param n := 5000; set I := 1..n; set A := setof{i in I, j in I: Uniform(0,1)<0.05 and i<>j} (i,j); param c{A} = Uniform(1,10); param cap{A} = Uniform(0,0.5); param supply{I} default 0; param demand{I} default 0; let supply[1] := 1; let supply[2] := 1; let demand[3] := 1; let demand[4] := 1; minimize Cost; node flowbal {i in I}: net_in = demand[i] - supply[i]; arc x{(i,j) in A} >= 0, <= cap[i,j], from flowbal[i], to flowbal[j], obj Cost c[i,j]; solve; display {(i,j) in A: x[i,j] > 0} (x[i,j]);

I actually prefer the equation based version: it conveys better what we are modeling. For more information about implementing network models in AMPL see [1].

#### GLPK

GLPK is an open source LP/MIP solver which includes a modeling tool called MathProg based on a subset of AMPL. Mathprog is different enough from AMPL such that the AMPL model did not work out of the box. I applied some edits things to make things work again:

param n := 5000; set I := 1..n; set A := setof{i in I, j in I: Uniform(0,1)<0.05 and i<>j} (i,j); set src := 1..2; set dest := 3..4; param c{A} := Uniform(1,10); param cap{A} := Uniform(0,0.5); param supply{i in I} := if i in src then 1 else 0; param demand{i in I} := if i in dest then 1 else 0; var x{(i,j) in A} >= 0, <= cap[i,j]; minimize Cost: sum{(i,j) in A} c[i,j]*x[i,j]; flowbal{i in I}: supply[i] + sum{(j,i) in A} x[j,i] = demand[i] + sum{(i,j) in A} x[i,j]; solve; display {(i,j) in A: x[i,j] > 0} (x[i,j]); end;

It is interesting to see how a free solver/modeling system would stack up against commercial ones. Unfortunately generating and solving this model took a very long time. Instead of less than a minute we deal with more than an hour:

D:\projects\blog\winglpk-4.65\glpk-4.65\w64>TimeMem-1.0.exe glpsol.exe --math network3.mod GLPSOL: GLPK LP/MIP Solver, v4.65 Parameter(s) specified in the command line: --math network3.mod Reading model section from network3.mod... 23 lines were read Generating Cost... Generating flowbal... Model has been successfully generated GLPK Simplex Optimizer, v4.65 5001 rows, 1250470 columns, 3751410 non-zeros Preprocessing... 5000 rows, 1250470 columns, 2500940 non-zeros Scaling... A: min|aij| = 1.000e+00 max|aij| = 1.000e+00 ratio = 1.000e+00 Problem data seem to be well scaled Constructing initial basis... Size of triangular part is 4999 0: obj = 6.939485101e-01 inf = 1.352e+01 (11) 120: obj = 2.066091319e+00 inf = 9.366e+00 (10) 239: obj = 5.857094094e+00 inf = 5.028e+00 (8) 360: obj = 5.857094094e+00 inf = 5.028e+00 (8) Perturbing LP to avoid stalling [403]... 479: obj = 5.857094094e+00 inf = 5.028e+00 (8) 597: obj = 5.532619386e+00 inf = 4.645e+00 (8) 716: obj = 5.532619386e+00 inf = 4.645e+00 (8) 835: obj = 1.004237246e+01 inf = 2.921e+00 (8) 953: obj = 1.025203418e+01 inf = 2.840e+00 (7) 1067: obj = 1.581285250e+01 inf = 1.285e+00 (5) 1192: obj = 1.805115142e+01 inf = 1.188e+00 (2) 1340: obj = 1.805115142e+01 inf = 1.188e+00 (2) 1503: obj = 2.212195621e+01 inf = 9.765e-01 (2) 1699: obj = 3.789698157e+01 inf = 0.000e+00 (0) * 1827: obj = 3.789698157e+01 inf = 0.000e+00 (473859) * 1955: obj = 3.789698157e+01 inf = 0.000e+00 (470365) * 2088: obj = 3.789698157e+01 inf = 0.000e+00 (459322) * 2221: obj = 3.789698157e+01 inf = 0.000e+00 (468268) * 2356: obj = 3.713937625e+01 inf = 0.000e+00 (456803) * 2488: obj = 3.713937625e+01 inf = 0.000e+00 (465825) * 2619: obj = 3.713937625e+01 inf = 0.000e+00 (475413) 1 * 2748: obj = 3.629538161e+01 inf = 0.000e+00 (429758) * 2879: obj = 3.586530520e+01 inf = 0.000e+00 (457580) * 3011: obj = 3.586530520e+01 inf = 0.000e+00 (466653) * 3150: obj = 3.586530520e+01 inf = 0.000e+00 (467626) * 3286: obj = 3.586530520e+01 inf = 0.000e+00 (469059) * 3423: obj = 3.586530520e+01 inf = 0.000e+00 (464494) * 3554: obj = 3.586530520e+01 inf = 0.000e+00 (470673) * 3683: obj = 3.586530520e+01 inf = 0.000e+00 (468476) * 3808: obj = 3.586530520e+01 inf = 0.000e+00 (458177) * 3934: obj = 3.568951686e+01 inf = 0.000e+00 (380721) 1 * 4065: obj = 3.568951686e+01 inf = 0.000e+00 (373842) * 4198: obj = 3.407645932e+01 inf = 0.000e+00 (454855) * 4331: obj = 3.407645932e+01 inf = 0.000e+00 (489226) * 4450: obj = 3.407645932e+01 inf = 0.000e+00 (478258) 1 * 4565: obj = 3.388831868e+01 inf = 0.000e+00 (463926) * 4687: obj = 3.200699481e+01 inf = 0.000e+00 (386710) * 4818: obj = 3.129121358e+01 inf = 0.000e+00 (431844) * 4947: obj = 3.120661156e+01 inf = 0.000e+00 (425111) 1 * 5081: obj = 3.120661156e+01 inf = 0.000e+00 (418646) * 5212: obj = 3.046484643e+01 inf = 0.000e+00 (449751) * 5337: obj = 2.989829501e+01 inf = 0.000e+00 (476458) * 5461: obj = 2.848322634e+01 inf = 0.000e+00 (474040) 1 * 5588: obj = 2.651301575e+01 inf = 0.000e+00 (363776) * 5717: obj = 2.576492847e+01 inf = 0.000e+00 (408484) * 5843: obj = 2.253496704e+01 inf = 0.000e+00 (423080) 1 * 5969: obj = 2.253496704e+01 inf = 0.000e+00 (403170) * 6099: obj = 2.215910899e+01 inf = 0.000e+00 (387070) * 6231: obj = 2.146461527e+01 inf = 0.000e+00 (401893) 1 * 6360: obj = 2.140440543e+01 inf = 0.000e+00 (420520) * 6485: obj = 2.140440543e+01 inf = 0.000e+00 (359102) * 6611: obj = 1.925278568e+01 inf = 0.000e+00 (417554) 1 * 6737: obj = 1.841509841e+01 inf = 0.000e+00 (426050) * 6860: obj = 1.581217012e+01 inf = 0.000e+00 (384451) 1 * 6990: obj = 1.406658394e+01 inf = 0.000e+00 (304416) * 7132: obj = 1.406658394e+01 inf = 0.000e+00 (312140) * 7269: obj = 1.406658394e+01 inf = 0.000e+00 (330591) * 7400: obj = 1.294736282e+01 inf = 0.000e+00 (335060) 1 * 7530: obj = 1.294736282e+01 inf = 0.000e+00 (361433) * 7667: obj = 1.256488191e+01 inf = 0.000e+00 (238709) 1 * 7802: obj = 1.256488191e+01 inf = 0.000e+00 (361552) 1 * 7938: obj = 1.256488191e+01 inf = 0.000e+00 (283082) * 8067: obj = 1.256488191e+01 inf = 0.000e+00 (339700) 1 * 8176: obj = 1.252840374e+01 inf = 0.000e+00 (248354) * 8318: obj = 1.252840374e+01 inf = 0.000e+00 (272086) 1 * 8458: obj = 1.252840374e+01 inf = 0.000e+00 (240051) * 8601: obj = 1.252840374e+01 inf = 0.000e+00 (203217) 1 * 8768: obj = 1.206036592e+01 inf = 0.000e+00 (68736) * 8936: obj = 1.194914307e+01 inf = 0.000e+00 (115418) 1 * 9092: obj = 1.186788602e+01 inf = 0.000e+00 (257936) 1 * 9257: obj = 1.186788602e+01 inf = 0.000e+00 (293133) * 9411: obj = 1.186788602e+01 inf = 0.000e+00 (190910) 1 * 9554: obj = 1.186788602e+01 inf = 0.000e+00 (232493) * 9703: obj = 1.162047323e+01 inf = 0.000e+00 (151447) 1 * 9852: obj = 1.162047323e+01 inf = 0.000e+00 (164831) * 10013: obj = 1.162047323e+01 inf = 0.000e+00 (160575) 1 * 10166: obj = 1.162047323e+01 inf = 0.000e+00 (277300) 1 * 10307: obj = 1.162047323e+01 inf = 0.000e+00 (236831) * 10454: obj = 1.162047323e+01 inf = 0.000e+00 (212227) 1 * 10604: obj = 1.162047323e+01 inf = 0.000e+00 (140137) * 10769: obj = 1.142233709e+01 inf = 0.000e+00 (98909) 1 * 10926: obj = 1.142233709e+01 inf = 0.000e+00 (241045) 1 * 11084: obj = 1.142233709e+01 inf = 0.000e+00 (181677) 1 * 11221: obj = 1.142233709e+01 inf = 0.000e+00 (230863) * 11373: obj = 1.142233709e+01 inf = 0.000e+00 (110799) 1 * 11542: obj = 1.142233709e+01 inf = 0.000e+00 (125143) 1 * 11710: obj = 1.142233709e+01 inf = 0.000e+00 (119827) 1 * 11870: obj = 1.142233709e+01 inf = 0.000e+00 (119131) * 12034: obj = 1.142233709e+01 inf = 0.000e+00 (109375) 1 * 12200: obj = 1.142233709e+01 inf = 0.000e+00 (101337) 1 * 12371: obj = 1.142233709e+01 inf = 0.000e+00 (107241) * 12542: obj = 1.142233709e+01 inf = 0.000e+00 (91039) 1 * 12719: obj = 1.142233709e+01 inf = 0.000e+00 (50343) 1 * 12875: obj = 1.142233709e+01 inf = 0.000e+00 (45627) * 13043: obj = 1.142233709e+01 inf = 0.000e+00 (61821) 1 * 13220: obj = 1.142233709e+01 inf = 0.000e+00 (56508) 1 * 13398: obj = 1.142233709e+01 inf = 0.000e+00 (63235) 1 * 13556: obj = 1.142233709e+01 inf = 0.000e+00 (171802) 1 * 13711: obj = 1.142233709e+01 inf = 0.000e+00 (178764) * 13870: obj = 1.142233709e+01 inf = 0.000e+00 (56535) 1 * 14046: obj = 1.142233709e+01 inf = 0.000e+00 (72873) 1 * 14219: obj = 1.142233709e+01 inf = 0.000e+00 (84913) 1 * 14385: obj = 1.142233709e+01 inf = 0.000e+00 (77343) * 14558: obj = 1.142233709e+01 inf = 0.000e+00 (62914) 1 * 14727: obj = 1.142233709e+01 inf = 0.000e+00 (48561) 1 * 14902: obj = 1.132754349e+01 inf = 0.000e+00 (84008) * 15054: obj = 1.132754349e+01 inf = 0.000e+00 (79216) 1 * 15216: obj = 1.132754349e+01 inf = 0.000e+00 (68852) 1 * 15386: obj = 1.119166284e+01 inf = 0.000e+00 (57191) 1 * 15556: obj = 1.119166284e+01 inf = 0.000e+00 (57834) * 15727: obj = 1.119166284e+01 inf = 0.000e+00 (75029) 1 * 15902: obj = 1.090002160e+01 inf = 0.000e+00 (58584) 1 * 16078: obj = 1.082030323e+01 inf = 0.000e+00 (48927) 1 * 16260: obj = 1.082030323e+01 inf = 0.000e+00 (38824) * 16439: obj = 1.080362034e+01 inf = 0.000e+00 (36321) 1 * 16626: obj = 1.042732846e+01 inf = 0.000e+00 (19686) 1 * 16812: obj = 1.042474771e+01 inf = 0.000e+00 (22113) 1 * 16996: obj = 1.042474771e+01 inf = 0.000e+00 (23719) * 17180: obj = 1.042474771e+01 inf = 0.000e+00 (24825) 1 * 17364: obj = 1.042474771e+01 inf = 0.000e+00 (20385) 1 * 17552: obj = 1.042474771e+01 inf = 0.000e+00 (19242) 1 * 17744: obj = 1.026837798e+01 inf = 0.000e+00 (18388) * 17930: obj = 1.026837798e+01 inf = 0.000e+00 (22530) 1 * 18120: obj = 1.025595159e+01 inf = 0.000e+00 (9025) 1 * 18296: obj = 1.020465344e+01 inf = 0.000e+00 (9133) 1 * 18463: obj = 1.020465344e+01 inf = 0.000e+00 (10438) 1 * 18621: obj = 1.015646186e+01 inf = 0.000e+00 (13266) * 18802: obj = 1.014150750e+01 inf = 0.000e+00 (19057) 1 * 18986: obj = 9.954708943e+00 inf = 0.000e+00 (7220) 1 * 19167: obj = 9.925813604e+00 inf = 0.000e+00 (7537) * 19351: obj = 9.900116820e+00 inf = 0.000e+00 (15139) 1 * 19538: obj = 9.863875257e+00 inf = 0.000e+00 (7051) 1 * 19727: obj = 9.861544899e+00 inf = 0.000e+00 (7586) 1 * 19921: obj = 9.861544899e+00 inf = 0.000e+00 (6291) * 20111: obj = 9.861544899e+00 inf = 0.000e+00 (5661) 1 * 20324: obj = 9.754583642e+00 inf = 0.000e+00 (3955) 1 * 20517: obj = 9.707349345e+00 inf = 0.000e+00 (1889) 1 * 20705: obj = 9.659064105e+00 inf = 0.000e+00 (1759) 1 * 20895: obj = 9.653194379e+00 inf = 0.000e+00 (1472) 1 * 21077: obj = 9.652094815e+00 inf = 0.000e+00 (1474) * 21271: obj = 9.650142953e+00 inf = 0.000e+00 (1149) 1 * 21465: obj = 9.630282332e+00 inf = 0.000e+00 (908) 1 * 21657: obj = 9.617880384e+00 inf = 0.000e+00 (448) 1 * 21846: obj = 9.606680468e+00 inf = 0.000e+00 (268) * 22042: obj = 9.606316377e+00 inf = 0.000e+00 (10) 1 Removing LP perturbation [22052]... * 22052: obj = 9.606316377e+00 inf = 0.000e+00 (0) OPTIMAL LP SOLUTION FOUND Time used: 728.3 secs Memory used: 2433.8 Mb (2552071521 bytes) Display statement at line 21 x[1,252].val = 0.149840161204338 x[1,379].val = 0.074746768688783 x[1,550].val = 0.0453869814518839 x[1,1084].val = 0.0445991435553879 x[1,1540].val = 0.0551238842308521 x[1,2345].val = 0.0476873791776597 x[1,2475].val = 0.00135596189647913 x[1,2749].val = 0.13588639209047 x[1,2884].val = 0.0230762232095003 x[1,3002].val = 0.00382565474137664 x[1,3188].val = 0.323973921360448 x[1,3645].val = 0.0407408073078841 x[1,4228].val = 0.0537567210849375 x[2,341].val = 0.386876111850142 x[2,847].val = 0.0894879563711584 x[2,1225].val = 0.0989777501672506 x[2,1259].val = 0.0705436542630196 x[2,1572].val = 0.0149974161759019 x[2,2406].val = 0.0154880676418543 x[2,2449].val = 0.00948114111088216 x[2,3169].val = 0.0226166676729918 x[2,3568].val = 0.0758920384105295 x[2,3570].val = 0.0437631316017359 x[2,3575].val = 0.020591959124431 x[2,4377].val = 0.00492054037749767 x[2,4548].val = 0.146363565232605 x[76,3].val = 0.149840161204338 x[170,4].val = 0.0154880676418543 x[252,76].val = 0.149840161204338 x[341,493].val = 0.31903103669174 x[341,1498].val = 0.0137350093573332 x[341,4015].val = 0.0541100658010691 x[379,1633].val = 0.0551570120733231 x[379,4157].val = 0.0195897566154599 x[493,2287].val = 0.31903103669174 x[502,1162].val = 0.020591959124431 x[550,2869].val = 0.0453869814518839 x[692,4].val = 0.0551570120733231 x[847,3240].val = 0.0894879563711584 x[917,3].val = 0.151284105610102 x[1007,3].val = 0.0541100658010691 x[1084,2428].val = 0.0425937338732183 x[1084,2673].val = 0.00200540968216956 x[1162,3].val = 0.020591959124431 x[1202,2963].val = 0.0551238842308521 x[1225,4900].val = 0.0989777501672506 x[1253,917].val = 0.00492054037749767 x[1259,2480].val = 0.0705436542630196 x[1344,3].val = 0.0537567210849375 x[1493,4].val = 0.165379994781688 x[1498,4].val = 0.0137350093573332 x[1540,1202].val = 0.0551238842308521 x[1572,2287].val = 0.0149974161759019 x[1633,692].val = 0.0551570120733231 x[1951,4].val = 0.121594417840242 x[2142,4].val = 0.157906056847423 x[2155,4].val = 0.0135977419558913 x[2158,1493].val = 0.0758920384105295 x[2271,4].val = 0.0230762232095003 x[2287,4].val = 0.334028452867642 x[2345,2142].val = 0.0220196647569537 x[2345,3479].val = 0.025667714420706 x[2373,3].val = 0.0437631316017359 x[2406,170].val = 0.0154880676418543 x[2428,3].val = 0.0425937338732183 x[2449,4892].val = 0.00948114111088216 x[2475,2793].val = 0.00135596189647913 x[2480,2675].val = 0.0705436542630196 x[2673,3].val = 0.00200540968216956 x[2675,4].val = 0.0705436542630196 x[2749,4340].val = 0.13588639209047 x[2793,3].val = 0.00135596189647913 x[2869,3].val = 0.0453869814518839 x[2884,2271].val = 0.0230762232095003 x[2963,3].val = 0.0551238842308521 x[3002,3772].val = 0.00382565474137664 x[3083,3].val = 0.0467328219674528 x[3169,1951].val = 0.0226166676729918 x[3188,3585].val = 0.323973921360448 x[3240,1493].val = 0.0894879563711584 x[3479,4].val = 0.025667714420706 x[3568,2158].val = 0.0758920384105295 x[3570,2373].val = 0.0437631316017359 x[3575,502].val = 0.020591959124431 x[3585,3].val = 0.323973921360448 x[3645,4183].val = 0.0135977419558913 x[3645,4619].val = 0.0271430653519928 x[3772,4].val = 0.00382565474137664 x[4015,1007].val = 0.0541100658010691 x[4157,3083].val = 0.0195897566154599 x[4183,2155].val = 0.0135977419558913 x[4228,1344].val = 0.0537567210849375 x[4340,2142].val = 0.13588639209047 x[4377,1253].val = 0.00492054037749767 x[4548,917].val = 0.146363565232605 x[4619,3083].val = 0.0271430653519928 x[4892,3].val = 0.00948114111088216 x[4900,1951].val = 0.0989777501672506 Model has been successfully processed Exit code : 0 Elapsed time : 4343.45 Kernel time : 11.16 (0.3%) User time : 4053.41 (93.3%) page fault # : 724488 Working set : 2579972 KB Paged pool : 86 KB Non-paged pool : 29 KB Page file size : 2665148 KB D:\projects\blog\winglpk-4.65\glpk-4.65\w64>

The solver takes about 730 seconds, and the modeling tool needs the rest of the 4340 seconds (i.e. about an hour). I suspect running through the sparse two-dimensional set

**A**in the

**flowbal**equation is the killer.

#### References

- Chapter 15, Network Linear Programs, AMPL book, https://ampl.com/BOOK/CHAPTERS/18-network.pdf

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