Case Study 48 Vehicle Routing ProblemVehicle Routing ProblemProblem DescriptionThevehicleroutingproblemiswell knownintheareaof combinatorial optimiation!Themain reason for such popularity is this problem has a number of applications! "or e#ample$the problem of scheduling the delivery of products to a customer using a fleet of vehicles canbe modeled as a vehicle routing problem! This problem is a generaliation of the travelingsalesman problem!The vehicle routing problem states the following% &Given% 'a( a fleet of K capacitated vehiclesdomiciled at a common depot$ say node )* 'b( a set of customer sites j + )$,$n each with aprescribed demand dj* and 'c( the costcijof traveling from location ito j*Findthe set ofroutesfordelivering'pickingup( thegoodstocustomersitesat theminimumcost! -eassume that the fleet of vehicles is homogeneous and that each vehicle has a capacity of uunits!. '/hu0a et al. ')112((!The aim of this pro0ect is to build a decision support system that will enable companies toidentifyaschedulefor deliveringproductstocustomersbysolvingthevehicleroutingproblem! 3elow we present an integer programming formulation of this problem! To learnmore about the vehicle routing problem$ we refer the students to /hu0a et al. ')112(!Integer Programming FormulationThe following notation is used%A the set of arcs of the networkK the total number of vehiclesu the capacity of a vehiclen the total number of nodescij the cost of using arc 'i$j( ''i$j(A(di the demand at node i!The decision variables are as follows%kijx takes the value ) if vehicle k travels from site i to site jyij takes the value ) if some vehicle travels from site i to site j!The ob0ective is to minimie the total cost of delivering the products to customers! The firstset of constraints shows that there is at most one vehicle traveling from site i to site j 'for all'i$j(A(! Thesecondand thirdsetsofconstraintsshowthat eachcustomersite isvisitede#actly once! / vehicle willenter and leave a city e#actly once! The fourth and fifth set ofconstraintsshowthat therewill bee#actlyKvehiclesleavingthedepot ande#actlyKvehicles entering the depot! The si#th set of constraints shows that vehicle capacity shouldnot be violated! The seventh set of constraints is the sub4tour elimination constraints! Theeighth set of constraints is the integrality constraints! 48Case Study 48 Vehicle Routing Problem) 8 ( . ,..., 1 , ) , ( } 1 , 0 { ,) 7 ( } ,..., 3 , 2 { 1) 6 ( , ,..., 1) 5 () 4 () 3 ( , ,..., 1 1) 2 ( , ,..., 1 1) 1 ( , ) , (:: min2 1111 11111 ) , (K k A j i x yn Q Q yK k u x dK yK yn j yn i yA j i y xto Subjectx ckij ijQ i Q jijninjkij iniinjjniijnjijijKkkijKk A j ikij ij= = === == = = = ======= Heuristic Solution Approaches 5earest 5eighbor% 6et V* denote the set of customer sites that have not been visited yet by a vehicle! )! Start at the depot 'node )( and construct a path consisting of nodes )$i)$ ,$i0$ ) where{ } ( ) * : min arg1V k c ik i jj =* ji i id d d u + + + ...2 1$ and any other node* V s is such thats i i id d d d uj+ + + + < ...2 1!7! Repeat Step ) until all the sites are visited! Thetotal number of pathsconstructedpresentsthetotal numberof vehiclesneededtodeliver the products to customers!Tolearnmoreabout thenearest neighbor heuristicaswell asother heuristicsolutionapproaches for the traveling salesman problem$ we refer the students to Chopra and 8eindl'799)(! Excel Spreadsheets)! 3uild a spreadsheet that presents the demand at each customer site!Case Study 48 Vehicle Routing Problem7! 3uild a spreadsheet that presents the cost of traveling from customer site i to site j! User Interface)! 3uild a welcome form!7! 3uild a data entry form! The following are suggestions to help you design this form! :nthisformincludetwooptionbuttons! Theoptionbuttonsenabletheuser toselectwhether to read the data from a file or manually enter the data! :nclude a commandbutton that$ when clicked on$ performs these actions% a! :f the user chose to read the data from a file$ a te#t bo# should appear where theuser types in the name of the file!b! :f the user chose to enter the data manually$ a te#t bo# appears where the user cantype in the total number of customer sites 'n(! ;pon submission of this information$two tables appear! The first table$ with dimensions ) by n$ allows the user to enterthe data about demand in each customer site! The second table$ with dimensions nby n$ enables the user to type in the cost of traveling from one site to the others!2! 3uild a form that presents the following details about a related e#ample! The user canunderstand how the nearest neighbor algorithm works by using this form!a! The problem statement! b! / graphical representation of the corresponding network!c! :nteger programming formulation!d! Present howthe nearest neighbor algorithmwas implemented to solve thise#ample! e! The optimal number of vehicles needed to deliver the products to customers andthe corresponding transportation costs!f! / graphical representation of the optimal solution 'vehicle tours(! 4! 3uild a form that allows the user to solve the problem and perform a sensitivity analysis!