Małgorzata Stera Mateusz Cicheński, Mateusz Jarus, Michał Miszkiewicz, Jarosław Szymczak Istitute of Computig Sciece Pozań Uiversity of Techology Pozań - Polad
Scope of the Talk Problem defiitio MP Formulatio Heuristic Algorithm Geetic Algorithm Computatioal Experimets Coclusios Malgorzata.Stera@cs.put.poza.pl 1/20
Problem Defiitio web service for charitable orgaizatios gatherig charitable orgaizatios ad doors allowig submittig requests ad registerig offers for various products maagig database for registered users supportig supply process with optimizatio algorithms Malgorzata.Stera@cs.put.poza.pl 2/20
Problem Defiitio charitable orgaizatio (customer) products are offered by depots (doors, warehouses, shops) i amout of a ij uits at price c ij uit trasportatio cost T demad for m types of products distace t ir i amout of d j uits positio of a depot i the route y ik amout of ordered uits x ij of particular products Malgorzata.Stera@cs.put.poza.pl 3/23
MP Formulatio mi x i1 m j1 ij c ij + T ( y i 1t 0i j1 i1 r1 1 k1 max{ 0,y y 1} t (y (1 y ))) ik r,k 1 ir ik i1 r1 r1 r,k 1 uder costraits i1 x ij x ij x ij a ij d j j 1...m i 1..., j 1...m 0 ad iteger i 1...,j order completio subproblem 1...m k1 k1 i1 i1 y ik order delivery subproblem y y y y ik ik ik i,k 1 mi{ 1, 1 1 {0,1} i1 m j1 x ij i 1... } k 1... y ik i 1... k 1... 1 i 1...,k 1... 1 Malgorzata.Stera@cs.put.poza.pl 4/20
Problem Formulatio selectig depots offerig demaded products at the lowest prices order completio easy problem greedy solutio is optimal determiig the shortest route to pick up products from these depots order delivery hard problem reduces to the shortest Hamiltoia cycle order completio ad delivery is strogly NP-hard as a variat of Travellig Purchaser Problem (Ramesh 1981) Malgorzata.Stera@cs.put.poza.pl 5/20
Heuristic Algorithm two-phase heuristic approach selectig depots costructig a tour Malgorzata.Stera@cs.put.poza.pl 6/20
Heuristic Algorithm selectig depots - choosig depots util demad is satisfied orderig depots accordig to product cost (greedy heuristic) weighted priorities (priority heuristic), based o: total distace to other locatios total cost of demaded products available at a depot total cost of all demaded products various priority weights result i various list heuristics ad various sets of selected depots Malgorzata.Stera@cs.put.poza.pl 7/20
Heuristic Algorithm costructig a tour from selected depots Miimum Spaig Tree Heuristic (Hedl & Karp 1970) costructig miimum spaig tree by Kruskal Algorithm traversig the tree accordig to Depth First Search Strategy covertig DFS sequece to the Hamiltoia cycle MSTH is 2-approximatio algorithm Malgorzata.Stera@cs.put.poza.pl 8/20
Bouds heuristic solutio determies upper boud (UB) lower boud LB= m ~ x, jc i i,j T( mi {t0i} j1 i1 i1... referece bouds RB= mi {t i0 }) m ~ x, jc i i,j T (mi{t0i} j1 i1 i1... i1... ~ t [k] mi {t i0 }) k1 i1... t [k] - k th distace betwee depots 1 Malgorzata.Stera@cs.put.poza.pl 9/20
Geetic Algorithm solutio is a sequece of assigmets: umber of product uits ordered from a depot oe product ca be take from more tha oe depot (to determie a complete solutio a tour is costructed by MST heuristic) iitial populatio heuristic solutios correspodig to various priorities weights radom solutios ew populatio replaces the previous oe Malgorzata.Stera@cs.put.poza.pl 10/20
Geetic Algorithm Operators oe-poit crossover ad two-poit crossover exchagig parts of assigmets product-depot ifeasible offsprig repair procedure: exceedig the product availability at a certai depot takig products from aother depot i offsprig takig products from a depot i paretal solutio mutatio replacig a give umber of assigmets (product-depot) with radom assigmet Malgorzata.Stera@cs.put.poza.pl 11/20
Geetic Algorithm - Selectio selectig matig populatio accordig to crossover rate roulette selectio (accordig to criterio values) rakig selectio (accordig to the positio i rakig) touramet selectio (the best solutio from radomly chose groups) mutatio accordig to mutatio rate Malgorzata.Stera@cs.put.poza.pl 12/20
Geetic Algorithm Termiatio Coditio the maximum umber of geeratios the maximum umber of geeratios without improvemet exceedig the satisfyig ratio of the criterio value improvemet Malgorzata.Stera@cs.put.poza.pl 13/20
Computatioal Experimets radom istaces reflectig real world scearios depots located i 48 Polish cities the charitable orgaizatio located i Pozań distaces correspod to road distaces (Big Maps) uit trasportatio cost determied by govermet regulatios Malgorzata.Stera@cs.put.poza.pl 14/20
Computatioal Experimets a sigle order cotais of 5 to 200 product types (from 1 to 10 uits of each type) prices of products i depots are determied based o basic price modified by discout factor geerated with ormal distributio [-50%, +50%] Malgorzata.Stera@cs.put.poza.pl 15/20
Computatioal Experimets availability of products i depots are geerated accordig to 3 scearios: roud robi distributio demaded uits of a product are placed i depots oe by oe all depots have to be visited (order delivery is crucial) cloe distributio all demaded products are available i all depots prices of products are crucial (order completio is crucial) eve distributio availability i all depots is icreased by oe uit at the time util demad is exceeded prices ad distaces are crucial Malgorzata.Stera@cs.put.poza.pl 16/20
Time per geeratio Malgorzata.Stera@cs.put.poza.pl 17/20
Trasportatio cost vs. Products cost Malgorzata.Stera@cs.put.poza.pl 18/20
Criterio value vs. lower boud Roud Robi distributio Cloe distributio Eve distributio Malgorzata.Stera@cs.put.poza.pl 19/20
Coclusios web service devoted for charitable istitutios data base of offered/doated products ad submitted requests optimizatio tool supportig realizig orders by miimizig products cost ad trasportatio cost solvig a variat of travellig purchaser problem by heuristic list algorithm geetic algorithm validatio algorithms i computatioal experimets solutio quality close to the lower boud short computatioal time acceptable by web service users Malgorzata.Stera@cs.put.poza.pl 20/20