## Vehicle Routing Problem (dcmpe08)

```
/***************************************************************/
/*                                                             */
/*            S A S   S A M P L E   L I B R A R Y              */
/*                                                             */
/*    NAME: dcmpe08                                            */
/*   TITLE: Vehicle Routing Problem (dcmpe08)                  */
/* PRODUCT: OR                                                 */
/*  SYSTEM: ALL                                                */
/*    KEYS: OR                                                 */
/*   PROCS: OPTMODEL, SGPLOT                                   */
/*    DATA:                                                    */
/*                                                             */
/* SUPPORT:                             UPDATE:                */
/*     REF:                                                    */
/*    MISC: Example 8 from the Decomposition Algorithm         */
/*          chapter of Mathematical Programming.               */
/*                                                             */
/***************************************************************/

/* 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
;

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);

/* 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;

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;

```