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Vehicle routing Description A transporter has to deliver heating oil from a refinery
to a certain number of clients. The distances between the
clients and the refinery and the demands in liters for
the different sites are given. The transport company uses
tankers with a capacity of 39000 liters for the deliveries.
Determine the tours for delivering to all clients that
minimize the total number of kilometers driven. Further explanation of this example:
'Applications of optimization with Xpress-MP',
Section 10.4 'Heating oil delivery'
Source Files Data Files vrp_graph.mos (!****************************************************** Mosel Example Problems ====================== file vrp.mos ```````````` TYPE: Vehicle routing problem (VRP) DIFFICULTY: 4 FEATURES: MIP problem, formulation of constraints to eliminate inadmissible subtours, definition of model cuts, selection with `|', algorithm for printing the tours, graphical solution representation DESCRIPTION: A transporter has to deliver heating oil from a refinery to a certain number of clients. The distances between the clients and the refinery and the demands in liters for the different sites are given. The transport company uses tankers with a capacity of 39000 liters for the deliveries. Determine the tours for delivering to all clients that minimize the total number of kilometers driven. FURTHER INFO: `Applications of optimization with Xpress-MP', Section 10.4 `Heating oil delivery' (c) 2008 Fair Isaac Corporation author: S. Heipcke, 2002, rev. Sep. 2017 *******************************************************!) model "Heating oil delivery" uses "mmxprs", "mmsvg" declarations NS = 7 SITES = 1..NS ! Set of locations, 1=refinery CLIENTS = 2..NS DEM: array(SITES) of integer ! Demands DIST: array(SITES,SITES) of integer ! Distances between locations CAP: integer ! Lorry capacity prec: array(SITES,SITES) of mpvar ! 1 if i immediately precedes j, ! 0 otherwise quant: array(CLIENTS) of mpvar ! Quantity delivered up to i end-declarations initializations from 'vrp.dat' DEM DIST CAP end-initializations ! Objective: total distance driven Length:= sum(i,j in SITES | i<>j) DIST(i,j)*prec(i,j) ! Enter and leave every city only once (except the depot) forall(j in CLIENTS) EnterOnce(j):= sum(i in SITES| i<>j) prec(i,j) = 1 forall(i in CLIENTS) LeaveOnce(i):= sum(j in SITES| i<>j) prec(i,j) = 1 ! If i is the first client of a tour, then quant(i)=DEM(i) forall(i in CLIENTS) QuantFirst(i):= quant(i) <= CAP + (DEM(i)-CAP)*prec(1,i) ! If j comes just after i in a tour, then quant(j) is greater than the ! quantity delivered during the tour up to i plus the quantity to be ! delivered at j (to avoid loops and keep capacity limit of the tanker) forall(i,j in CLIENTS| i<>j) QuantSucc(i,j):= quant(j) >= quant(i) + DEM(j) - CAP + CAP*prec(i,j) + (CAP-DEM(j)-DEM(i))*prec(j,i) ! Additional constraints: ! If i is not the first client of a tour, quant(i) is larger than the sum ! of the quantities to deliver to i and to his predecessor on the tour (! declarations modcut: array(CLIENTS) of linctr end-declarations forall(i in CLIENTS) do modcut(i):= quant(i) >= DEM(i) + sum(j in SITES| i<>j) DEM(j)*prec(j,i) setmodcut(modcut(i)) end-do !) forall(i in CLIENTS) do quant(i) <= CAP quant(i) >= DEM(i) end-do forall(i,j in SITES | i<>j) prec(i,j) is_binary ! Uncomment the following line to see the Optimizer log ! setparam("XPRS_VERBOSE",true) ! Solve the problem minimize(Length) ! Solution printing writeln("Total distance: ", getobjval) forall(i in CLIENTS) if(getsol(prec(1,i))>0) then ct:=DEM(i) writeln(1, " -> ", i) p:=i while(p<>1) do n:= integer(round(sum(j in SITES) j*getsol(prec(p,j)))) writeln(p, " -> ", n) ct+=DEM(n) p:=n end-do writeln("Quantity delivered: ", ct) end-if ! Solution drawing declarations X,Y: array(SITES) of integer ! x-y-coordinates of sites end-declarations initializations from 'vrp.dat' [X,Y] as 'POS' end-initializations minx:=min(i in SITES)X(i)-15; miny:=min(i in SITES)Y(i)-15 svgsetgraphviewbox(0, 0, max(i in SITES)X(i)+15, max(i in SITES)Y(i)+25) svgsetgraphscale(2) svgsetgraphpointsize(2) svgaddgroup("Sites", "Clients", SVG_BLACK) forall(i in CLIENTS) do svgaddpoint(X(i), Y(i)) svgaddtext(X(i)+0.5, Y(i)+1, text(i-1)) end-do svgaddgroup("Ref", "Refinery", SVG_BROWN) svgaddpoint(X(1), Y(1)) svgaddtext(X(1)+0.5, Y(1)-5, "Refinery") svgaddgroup("Routes", "Delivery routes") forall(i in CLIENTS) if(getsol(prec(1,i))>0) then svgaddarrow(X(1), Y(1), X(i), Y(i)) p:=i while(p<>1) do n:= integer(round(sum(j in SITES) j*getsol(prec(p,j)))) svgaddarrow(X(p), Y(p), X(n), Y(n)) p:=n end-do end-if svgsave("vrp.svg") svgrefresh svgwaitclose("Close browser window to terminate model execution.", 1) end-model | |||||||||

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