The Decomposition Algorithm

The vehicle routing problem (VRP) finds a minimum-cost routing of a fixed number of vehicles to service the demands of a set of customers. Define a set of customers, and a demand, , for each customer c. Let be the set of nodes, including the vehicle depot, which are designated as node . Let be the set of arcs, be the set of vehicles (each of which has capacity ), and be the travel time from node i to node j.

Let be a binary variable that, if set to 1, indicates that node i is visited by vehicle k. Let be a binary variable that, if set to 1, indicates that arc is traversed by vehicle k, and let be a continuous variable that denotes the amount of product (flow) on arc that is carried by vehicle k.

A VRP can be formulated as a MILP as follows:

In this formulation, the Assignment constraints ensure that each customer is serviced by at least one vehicle. The objective function ensures that there exists an optimal solution that never assigns a customer to more than one vehicle. The LeaveNode and EnterNode constraints enforce the condition that if node i is visited by vehicle k, then vehicle k must use exactly one arc that enters node i and one arc that leaves node i. Conversely, if node i is not visited by vehicle k, then no arcs that enter or leave node i can be used by vehicle k. The FlowBalance constraints define flow conservation at each node for each vehicle. That is, if a node i is visited by vehicle k, then the amount of product from vehicle k that enters and leaves that node must equal the demand at that node. Conversely, if node i is not visited by vehicle k, then the amount of product from vehicle k that enters and leaves that node must be 0. The VehicleCapacity constraints enforce the condition that the amount of product in each vehicle must always be less than or equal to the vehicle capacity . Finally, the Depot constraints enforce the condition that each vehicle must start and end at the depot node.

In this formulation, the vehicle identifier is arbitrary. Consider a decomposition by vehicle, where the Assignment constraints form the master problem and all other constraints form identical routing subproblems. As described in the section Special Case: Identical Blocks and Ryan-Foster Branching, this is a situation in which an aggregate formulation can greatly improve performance by reducing symmetry. Because you know that there exists an optimal solution that satisfies the master Assignment constraints at equality, you can force the use of Ryan-Foster branching by specifying the option VARSEL=RYANFOSTER.

VRPLIB, located at http://www.coin-or.org/SYMPHONY/branchandcut/VRP/data/index.htm, is a set of benchmark instances of the VRP. The following data set, vrpdata, represents an instance from VRPLIB that has 22 nodes and eight vehicles (P-n22-k8.vrp), which was originally described in Augerat et al. (1995). The data set lists each node, its coordinates, and its demand.

/* number of vehicles available */
%let num_vehicles = 8;
/* capacity of each vehicle */
%let capacity = 3000;
/* node, x coordinate, y coordinate, demand */
data vrpdata;
   input node x y demand;
   datalines;
1  145 215    0
2  151 264 1100
3  159 261  700
4  130 254  800
5  128 252 1400
6  163 247 2100
7  146 246  400
8  161 242  800
9  142 239  100
10 163 236  500
11 148 232  600
12 128 231 1200
13 156 217 1300
14 129 214 1300
15 146 208  300
16 164 208  900
17 141 206 2100
18 147 193 1000
19 164 193  900
20 129 189 2500
21 155 185 1800
22 139 182  700
;

The following PROC OPTMODEL statements read in the data, declare the optimization model, and use the decomposition algorithm to solve it:

proc optmodel;
   /* read the node location and demand data */
   set NODES;
   num x {NODES};
   num y {NODES};
   num demand {NODES};
   num capacity = &capacity;
   num num_vehicles = &num_vehicles;
   read data vrpdata into NODES=[node] x y demand;
   set ARCS = {i in NODES, j in NODES: i ne j};
   set VEHICLES = 1..num_vehicles;

   /* define the depot as node 1 */
   num depot = 1;

   /* define the arc cost as the rounded Euclidean distance */
   num cost {<i,j> in ARCS} = round(sqrt((x[i]-x[j])^2 + (y[i]-y[j])^2));

   /* Flow[i,j,k] is the amount of demand carried on arc (i,j) by vehicle k */
   var Flow {ARCS, VEHICLES} >= 0 <= capacity;
   /* UseNode[i,k] = 1, if and only if node i is serviced by vehicle k */
   var UseNode {NODES, VEHICLES} binary;
   /* UseArc[i,j,k] = 1, if and only if arc (i,j) is traversed by vehicle k */
   var UseArc {ARCS, VEHICLES} binary;

   /* minimize the total distance traversed */
   min TotalCost = sum {<i,j> in ARCS, k in VEHICLES} cost[i,j] * UseArc[i,j,k];

   /* each non-depot node must be serviced by at least one vehicle */
   con Assignment {i in NODES diff {depot}}:
      sum {k in VEHICLES} UseNode[i,k] >= 1;

   /* each vehicle must start at the depot node */
   for{k in VEHICLES} fix UseNode[depot,k] = 1;

   /* some vehicle k traverses an arc that leaves node i
      if and only if UseNode[i,k] = 1 */
   con LeaveNode {i in NODES, k in VEHICLES}:
      sum {<(i),j> in ARCS} UseArc[i,j,k] = UseNode[i,k];

   /* some vehicle k traverses an arc that enters node i
      if and only if UseNode[i,k] = 1 */
   con EnterNode {i in NODES, k in VEHICLES}:
      sum {<j,(i)> in ARCS} UseArc[j,i,k] = UseNode[i,k];

   /* the amount of demand supplied by vehicle k to node i must equal demand
      if UseNode[i,k] = 1; otherwise, it must equal 0 */
   con FlowBalance {i in NODES diff {depot}, k in VEHICLES}:
       sum {<j,(i)> in ARCS} Flow[j,i,k] - sum {<(i),j> in ARCS} Flow[i,j,k]
       = demand[i] * UseNode[i,k];

   /* if UseArc[i,j,k] = 1, then the flow on arc (i,j) must be at most capacity
      if UseArc[i,j,k] = 0, then no flow is allowed on arc (i,j) */
   con VehicleCapacity {<i,j> in ARCS, k in VEHICLES}:
      Flow[i,j,k] <= Flow[i,j,k].ub * UseArc[i,j,k];

   /* decomp by vehicle */
   for {i in NODES, k in VEHICLES} do;
      LeaveNode[i,k].block = k;
      EnterNode[i,k].block = k;
   end;
   for {i in NODES diff {depot}, k in VEHICLES} FlowBalance[i,k].block = k;
   for {<i,j> in ARCS, k in VEHICLES} VehicleCapacity[i,j,k].block = k;

   /* solve using decomp (aggregate formulation) */
   solve with MILP / varsel=ryanfoster decomp=(logfreq=20);

The following OPTMODEL statements create node and edge data for the optimal routing:

   /* create solution data set */
   str color {k in VEHICLES} =
      ['red' 'green' 'blue' 'black' 'orange' 'gray' 'maroon' 'purple'];
   create data node_data from [i] x y;
   create data edge_data from [i j k]=
      {<i,j> in ARCS, k in VEHICLES: UseArc[i,j,k].sol > 0.5}
      x1=x[i] y1=y[i] x2=x[j] y2=y[j] linecolor=color[k];
quit;

The solution summary is displayed in Output 15.8.1.

The iteration log is displayed in Output 15.8.2.

The following DATA step and call to PROC SGPLOT generate a plot of the optimal routing. The plot is displayed in Output 15.8.3.

data sganno(drop=i j);
   retain drawspace "datavalue" linethickness 1;
   set edge_data;
   function = 'line';
run;

proc sgplot data=node_data sganno=sganno;
   scatter x=x y=y / datalabel=i;
   xaxis display=none;
   yaxis display=none;
run;

Output 15.8.3: Optimal Routing