```MODEL: ! The Vehicle Routing Problem (VRP); !************************************; ! WARNING: Runtimes for this model ; ! increase dramatically as the number; ! of cities increase. Formulations ; ! with more than a dozen cities ; ! WILL NOT SOLVE in a reasonable ; ! amount of time ; !************************************; SETS: ! Q(I) is the amount required at city I, U(I) is the accumulated delivers at city I ; CITY/1..8/: Q, U; ! DIST(I,J) is the distance from city I to city J X(I,J) is 0-1 variable: It is 1 if some vehicle travels from city I to J, 0 if none; CXC( CITY, CITY): DIST, X; ENDSETS DATA: ! city 1 represent the common depo; Q = 0 6 3 7 7 18 4 5; ! distance from city I to city J is same from city J to city I distance from city I to the depot is 0, since the vehicle has to return to the depot; DIST = ! To City; ! Chi Den Frsn Hous KC LA Oakl Anah From; 0 996 2162 1067 499 2054 2134 2050!Chicago; 0 0 1167 1019 596 1059 1227 1055!Denver; 0 1167 0 1747 1723 214 168 250!Fresno; 0 1019 1747 0 710 1538 1904 1528!Houston; 0 596 1723 710 0 1589 1827 1579!K. City; 0 1059 214 1538 1589 0 371 36!L. A.; 0 1227 168 1904 1827 371 0 407!Oakland; 0 1055 250 1528 1579 36 407 0;!Anaheim; ! VCAP is the capacity of a vehicle ; VCAP = 18; ENDDATA ! Minimize total travel distance; MIN = @SUM( CXC: DIST * X); ! For each city, except depot....; @FOR( CITY( K)| K #GT# 1: ! a vehicle does not traval inside itself,...; X( K, K) = 0; ! a vehicle must enter it,... ; @SUM( CITY( I)| I #NE# K #AND# ( I #EQ# 1 #OR# Q( I) + Q( K) #LE# VCAP): X( I, K)) = 1; ! a vehicle must leave it after service ; @SUM( CITY( J)| J #NE# K #AND# ( J #EQ# 1 #OR# Q( J) + Q( K) #LE# VCAP): X( K, J)) = 1; ! U( K) is at least amount needed at K but can't exceed capacity; @BND( Q( K), U( K), VCAP); ! If K follows I, then can bound U( K) - U( I); @FOR( CITY( I)| I #NE# K #AND# I #NE# 1: U( K) >= U( I) + Q( K) - VCAP + VCAP * ( X( K, I) + X( I, K)) - ( Q( K) + Q( I)) * X( K, I); ); ! If K is 1st stop, then U( K) = Q( K); U( K) <= VCAP - ( VCAP - Q( K)) * X( 1, K); ! If K is not 1st stop...; U( K)>= Q( K)+ @SUM( CITY( I)| I #GT# 1: Q( I) * X( I, K)); ); ! Make the X's binary; @FOR( CXC: @BIN( X)); ! Minimum no. vehicles required, fractional and rounded; VEHCLF = @SUM( CITY( I)| I #GT# 1: Q( I))/ VCAP; VEHCLR = VEHCLF + 1.999 - @WRAP( VEHCLF - .001, 1); ! Must send enough vehicles out of depot; @SUM( CITY( J)| J #GT# 1: X( 1, J)) >= VEHCLR; END ```