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mathematical optimization - Formulating a tailor-made Maximum Flow problem in a directed graph


Let a digraph, G, be given, where t[i,j] and a[j,i], respectively, denote transfer capacity of i allocated for transferring its information to j and absorption capacity of j allocated for absorbing information coming from i. The term, x[i], defines the information holding capacity of vertex i. If the amount of information inflowing to vertex i is larger than its capacity, then vertex i will refuse the extra information.


The following code generates an example digraph G with n=6:



Clear[trs, abs, info, edgeCapMat, system1, reducedSystem1, sa, wG1];
SeedRandom[14];
n = 6;
d = 0.3;
G1 = RandomGraph[{Round[n], Round[n*(n - 1)*d]}, 
   DirectedEdges -> True];

trs = Table[ Table[t[i, j], {j, 1, n}], {i, 1, n}];
abs = Table[ Table[a[i, j], {j, 1, n}], {i, 1, n}] // Transpose;
info = Table[x[i], {i, 1, n}];

infoStocks[stock_] := DiagonalMatrix[stock];
edgeCapMat[trsCap_, absCap_] := (trsCap*absCap) - 
   DiagonalMatrix[Diagonal[trsCap*absCap]]; (* @JMissomewhatokay's contribution *)
system1 = infoStocks[info].edgeCapMat[trs, abs];

reducedSystem1 = 
 AdjacencyMatrix[G1]*
  system1;  (* the system associated with AdjacencyGraph "G" *)
sa = SparseArray[reducedSystem1];
wG1 = Graph[sa["NonzeroPositions"], EdgeWeight -> sa["NonzeroValues"],

   DirectedEdges -> True, VertexLabels -> "Name" , 
  EdgeLabels -> "EdgeWeight"]; (* symbolic weighted-G1 *)

(* @kglr's contribution  *)
ClearAll[edgeW];
edgeW = Module[{g = #, 
     e = DirectedEdge @@@ Partition[#, 2, 1] & /@ 
       FindPath[##, \[Infinity], All]}, 
    Transpose[{e, PropertyValue[{g, #}, EdgeWeight] & /@ # & /@ e}]] &;


edgeW[wG1, 5, 2]
HighlightGraph[wG1, edgeW[wG1, 5, 2][[All, 1]]]

enter image description here


(Note: Mathematica Code to produce G has been developed by @kglr (many of us know @kglr from his contributions in MSE):


One can formulate a Maximum Flow problem for each path in G. Below I present an example max-flow problem from vertex 5 to vertex 2. (This is only one of the 30 potential problems with n=6.)


To understand what I am doing, one may simply follow the digraph G given above. Some of the vertices and parameters are not required for the paths from 5 to 2. For example, t[2]==0 and a[5]==0 are irrelevant by construction, and vertex 6, not involved in any of the paths concerned, should be excluded from the following specific problem formulation.


Clear[objFn, constraintsAll, choiceVars, fc1, fc2, fc3, fc4, fc5, fc6, fct1, fct2, fct3, fct4, fct5, fca1, fca2, fca3, fca4, fca5, bc1, bc2, bc3, bc4, tc1, tc2, tc3, tc4, ac1, ac2, ac3, ac4];

(* Given *)

{x[1] = 5,   x[2] = 9,   x[3] = 3,   x[4] = 6,   x[5] = 7,   x[6] = 2};
{t[1] = 0.7, t[2] = 0,   t[3] = 0.8, t[4] = 0.3, t[5] = 0.8, t[6] = 0.4};
{a[1] = 0.9, a[2] = 0.7, a[3] = 0.6, a[4] = 0.7, a[5] = 0,   a[6] = 0.7};

Maximize ObjFn = t[5,3] a[3,5] x[5]
subject to:

(* feasibility constraints for: vertex information stock capacity *)
fc1 = t[5, 3] a[3, 5] x[5] <= x[3];
fc2 = t[3, 4] a[4, 3] x[3] <= x[4];

fc3 = t[3, 1] a[1, 3] x[3] <= x[1];
fc4 = t[4, 2] a[2, 4] x[4] <= x[2];  (* NEW: to be added *)
fc5 = t[1, 2] a[2, 1] x[1] <= x[2];  (* NEW: to be added *)
fc6 = t[4, 2] a[2, 4] x[4] + t[1, 2] a[2, 1] x[1] <= x[2]; (* NEW: to be added *)

(* feasibility constraints for: attention allocation *)
fct1 = 0 <= t[5, 3] <= 1;
fct2 = 0 <= t[3, 4] <= 1;
fct3 = 0 <= t[3, 1] <= 1;
fct4 = 0 <= t[4, 2] <= 1;

fct5 = 0 <= t[1, 2] <= 1;

fca1 = 0 <= a[3, 5] <= 1;
fca2 = 0 <= a[4, 3] <= 1;
fca3 = 0 <= a[1, 3] <= 1;
fca4 = 0 <= a[2, 4] <= 1;
fca5 = 0 <= a[2, 1] <= 1;

(* flow balancing conditions: inflow into i = outflow from i *)
bc1 = t[5, 3] a[3, 5] x[5] == (t[3, 4] a[4, 3] + t[3, 1] a[1, 3]) x[3]; (* inflow from 5 to 3 = outflow from 3 to 4 and 1 *)

bc2 = t[3, 4] a[4, 3] x[3] == t[4, 2] a[2, 4] x[4]; (* inflow from 3 to 4 = outflow from 4 to 2 *)
bc3 = t[3, 1] a[1, 3] x[3] == t[1, 2] a[2, 1] x[1]; (* NEW to be added: inflow from 3 to 1 = outflow from 1 to 2 *)
bc4 = t[4, 2] a[2, 4] x[4] + t[1, 2] a[2, 1] x[1] == t[5, 3] a[3, 5] x[5]; (* NEW to be added: total inflow to 2 = total outflow from 5 *)

(* transfer attention allocation constraints *)
tc1 = t[5, 3] <= t[5]; (* v5 allocates its total transfer attention to v3 *)
tc2 = t[3, 4] + t[3, 1] <= t[3]; (* v3 allocates its total transfer attention to v4 and v1 *)
tc3 = t[4, 2] <= t[4];
tc4 = t[1, 2] <= t[1];


(* absorption attention allocation constraints *)
ac1 = a[3, 5] <= a[3]; (* choice variable of v3 because v3 decides how much attention it should allocate for receiving information from v5 *)
ac2 = a[4, 3] <= a[4]; (* the same argument as above *)
ac3 = a[1, 3] <= a[1];
ac4 = a[2, 4] + a[2, 1] <= a[2]; (* v2 allocates its total attention b/w v4 and v1 *)

(* list of constraints *)
constraintsAll = {fc1, fc2, fc3, fc4, fc5, fc6, fct1, fct2, fct3, fct4, fct5, fca1, fca2, fca3, fca4, fca5, bc1, bc2, bc3, bc4, tc1, tc2, tc3, tc4, ac1, ac2, ac3, ac4};

(* list of choice variables: find the optimal values of these vars *)

choiceVars = {t[5, 3], t[3, 4], t[3, 1], t[4, 2], t[1, 2], a[3, 5], 
a[4, 3], a[1, 3], a[2, 4], a[2, 1]}

Maximize[{objFn, constraintsAll}, choiceVars]

The solution:


{2.16, {t[5, 3] -> 0.786664, t[3, 4] -> 0., t[3, 1] -> 0.8, t[4, 2]->0.278239, 
t[1, 2] -> 0.671632, a[3, 5] -> 0.392253, a[4, 3] -> 4.26136*10^-10, 
a[1, 3] -> 0.9, a[2, 4] -> 0., a[2, 1] -> 0.643209}}


Note: The above max-flow is only for the paths from 5 to 2. I want to solve the same problem for each one of the existing paths in G, and each problem should give me a set of values for {t[i,j], a[j,i]} that maximize the relevant objective function. Needless to say, each problem will have its own specific constraints and objective function to maximize.


Question


How can I automate the above problem formulation to find the system-wide optimal values for each path, i.e., {t[i,j], a[j,i] for all i and j}?.


(Note: This question is a simplified version of a question asked in MSE before. I made some improvement in the formulation.)



Answer



First added x[i]s as VertexCapacitys to wG1 so that we can later extract all the model elements from wG1:


wG1 = Graph[sa["NonzeroPositions"], EdgeWeight -> sa["NonzeroValues"],
   DirectedEdges -> True, VertexCapacity -> {i_ :> x[i]}, 
   VertexSize -> .2,  EdgeLabels ->  "EdgeWeight"];
SetProperty[wG1, VertexLabels -> 

    {i_ :>  Placed[PropertyValue[{wG1, i}, VertexCapacity], Center]}]; 

HighlightGraph[wG1, edgeW[wG1, 5, 2][[All, 1]]]

enter image description here


All functions below take three arguments, a graph, a source vertex and a sink vertex:


ClearAll[decisionVarsF, subGraphF, feasibleF, vInfoStockCapF, attentionAllocationF, 
 flowBalanceF, allConstraintsF , modelF, instanceF]

subGraphF =   Module[{el = edgeW[##][[All, 1]]}, 

    Graph[DeleteDuplicates[Flatten@el] , 
         VertexCapacity -> {v_ :> PropertyValue[{#, v}, VertexCapacity]}, 
         EdgeWeight -> {e_ :> PropertyValue[{#, e}, EdgeWeight]}]] &;
decisionVarsF  = Union@Flatten[EdgeList[subGraphF[##]] /. 
      DirectedEdge[v1_, v2_] :> {t[v1, v2], a[v2, v1]}] & ;
feasibleF = Thread[0 <= decisionVarsF [##] <= 1] &; 
flowBalanceF = Module[{g = subGraphF[##]}, 
    Equal @@ (Total[PropertyValue[{g, #}, EdgeWeight] & /@ 
       Cases[EdgeList[g], #] ] & /@ {DirectedEdge[#, _] , 
          Reverse[DirectedEdge[#, _] ]}) & /@  Most[Rest[VertexList@g]] ] & ;

 attentionAllocationF = Module[{gb = GatherBy[EdgeList[ subGraphF[##]] /. 
    DirectedEdge[v1_, v2_] :> Sequence[{v1, t[v1, v2]}, {v2, a[v2, v1]}], 
      {First, Head@#[[2]] &}] },
   Flatten[Replace[gb, p : {{_, (_a | _t)} ..} :> 
    Equal[(Head[p[[1, 2]]] /. {a -> α, t -> τ})[p[[1, 1]]], 
      Total[p[[All, 2]]]], ∞], 1]]&;  
vInfoStockCapF = Module[{g = subGraphF[##]},  
   Mean[#[[All, 1]]] <= Mean[#[[All, 2]]] & /@ 
     GatherBy[PropertyValue[{g, #}, EdgeWeight] <= 
        PropertyValue[{g, #[[2]]}, VertexCapacity] & /@ 

          EdgeList@subGraphF[##], Last]] &;
allConstraintsF = Flatten[Through[{feasibleF, vInfoStockCapF, attentionAllocationF, 
   flowBalanceF}@##]] &; 
objective = t[5, 3] a[3, 5] x[5];
modelF = {{objective, And @@ allConstraintsF[##]},  decisionVarsF[##]} &;
instanceF[params_] := modelF[##] /. Flatten[Thread[#[[1]] -> #[[2]]] & /@ 
  Thread[Array[#, 5] & /@ {x, τ, α} -> params]] &;

Examples:


modelF[wG1, 5, 2]



{{a[3, 5] t[5, 3] x[5],
0 <= a[1, 3] <= 1 && 0 <= a[2, 1] <= 1 && 0 <= a[2, 4] <= 1 && 0 <= a[3, 5] <= 1 && 0 <= a[4, 3] <= 1 && 0 <= t[1, 2] <= 1 && 0 <= t[3, 1] <= 1 && 0 <= t[3, 4] <= 1 && 0 <= t[4, 2] <= 1 && 0 <= t[5, 3] <= 1 &&
a[3, 5] t[5, 3] x[5] <= x[3] &&
a[4, 3] t[3, 4] x[3] <= x[4] &&
1/2 (a[2, 1] t[1, 2] x1 + a[2, 4] t[4, 2] x[4]) <= x[2] &&
a[1, 3] t[3, 1] x[3] <= x1 &&
Ï„[5] == t[5, 3] &&
α[3] == a[3, 5] &&

Ï„[3] == t[3, 1] + t[3, 4] &&
α[4] == a[4, 3] &&
τ[4] == t[4, 2] && α[2] == a[2, 1] + a[2, 4] &&
α1 == a[1, 3] &&
Ï„1 == t[1, 2] &&
a[1, 3] t[3, 1] x[3] + a[4, 3] t[3, 4] x[3] == a[3, 5] t[5, 3] x[5] &&
a[2, 4] t[4, 2] x[4] == a[4, 3] t[3, 4] x[3] &&
0 == a[2, 1] t[1, 2] x1 + a[2, 4] t[4, 2] x[4]},
{a[1, 3], a[2, 1], a[2, 4], a[3, 5], a[4, 3], t[1, 2], t[3, 1], t[3, 4], t[4, 2], t[5, 3]}}




parameters = {xx, tt, aa} = {{1, 3, 3, 15, 3}, {.7, 0, .8, .3, .8}, {.9, .7, .6, .7, 0}};

instanceF[parameters][wG1, 5, 2]


{{3 a[3, 5] t[5, 3],
0 <= a[1, 3] <= 1 && 0 <= a[2, 1] <= 1 && 0 <= a[2, 4] <= 1 && 0 <= a[3, 5] <= 1 && 0 <= a[4, 3] <= 1 && 0 <= t[1, 2] <= 1 && 0 <= t[3, 1] <= 1 && 0 <= t[3, 4] <= 1 && 0 <= t[4, 2] <= 1 && 0 <= t[5, 3] <= 1 &&
3 a[3, 5] t[5, 3] <= 3 &&
3 a[4, 3] t[3, 4] <= 15 &&
1/2 (a[2, 1] t[1, 2] + 15 a[2, 4] t[4, 2]) <= 3 &&

3 a[1, 3] t[3, 1] <= 1 &&
0.8 == t[5, 3] &&
0.6 == a[3, 5] &&
0.8 == t[3, 1] + t[3, 4] &&
0.7 == a[4, 3] &&
0.3 == t[4, 2] &&
0.7 == a[2, 1] + a[2, 4] &&
0.9 == a[1, 3] &&
0.7 == t[1, 2] &&
3 a[1, 3] t[3, 1] + 3 a[4, 3] t[3, 4] == 3 a[3, 5] t[5, 3] &&

15 a[2, 4] t[4, 2] == 3 a[4, 3] t[3, 4] &&
0 == a[2, 1] t[1, 2] + 15 a[2, 4] t[4, 2]},
{a[1, 3], a[2, 1], a[2, 4], a[3, 5], a[4, 3], t[1, 2], t[3, 1], t[3, 4], t[4, 2], t[5, 3]}}



NMaximize @@  instanceF[parameters][wG1,5,2]


NMaximize::nosat: Obtained solution does not satisfy the following constraints within Tolerance -> 0.001: {3 a[4,3] t[3,4]-15 a[2,4] t[4,2]==0,-a[2,1] t[1,2]-15 a[2,4] t[4,2]==0,3 a[1,3] t[3,1]+3 a[4,3] t[3,4]-3 a[3,5] t[5,3]==0}.


{1.44, {a[1, 3] -> 0.9, a[2, 1] -> 0.636639, a[2, 4] -> 0.0633612, a[3, 5] -> 0.6, a[4, 3] -> 0.7, t[1, 2] -> 0.7, t[3, 1] -> 0.37037, t[3, 4] -> 0.42963, t[4, 2] -> 0.3, t[5, 3] -> 0.8}}




Note: if we replace Equal with GreaterEqual in definition of attentionAllocationF above then we get a solution without infeasibility warning:


solution  = NMaximize @@  instanceF[parameters][wG1,5,2]


{1., {a[1, 3] -> 0.580483, a[2, 1] -> 2.82245*10^-12, a[2, 4] -> 0.0553563, a[3, 5] -> 0.598288, a[4, 3] -> 1.01812*10^-12, t[1, 2] -> 0.117302, t[3, 1] -> 0.574234, t[3, 4] -> 0.224126, t[4, 2] -> 5.48145*10^-13, t[5, 3] -> 0.557145}}



At this solution none of the attention allocation constraints is binding:


attentionAllocationF[wG1, 5, 2] /. GreaterEqual -> Equal /. 
  Flatten[Thread[#[[1]] -> #[[2]]] & /@ 
   Thread[Array[#, 5] & /@ {x, τ, α} -> parameters]]  /. solution[[2]]



{False, False, False, False, False, False, False, False}



Another example with vertex 1 as the sink:


NMaximize @@ instanceF[parameters][wG1, 5, 1]


 {1.44, {a[1, 2] -> 0.54086, a[1, 3] -> 0.0226264, a[2, 4] -> 0.474469,    a[3, 5] -> 0.6, a[4, 3] -> 0.645526, t[2, 1] -> 0.,   t[3, 1] -> 0.00494044, t[3, 4] -> 0.743407, t[4, 2] -> 0.202284,   t[5, 3] -> 0.8}}




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