Picture 1: The number of combinations. c = (n-1) (n-2) (n-3) = (n-1)! Formula 1: The number of combinations

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1 Optimizatio of logistic routes usig mathematical models Ig. Josef Košťálek Supervisor: Kava Michal, doc. Ig. CSc. Abstrakt Logistika je oborem acházející silé uplatěí v průmyslové výrobě i ávazých oblastech. Logistika jako obor využívá pozatky celé řady jiých disciplí. Předmětem mého příspěvku je ukázat využití ěkterých speciálích matematických a výpočetích aplikací v oblasti staoveí ejvhodější logistické trasy. Představuji zde mou vytvořeý model ve sado dostupém prostředí MS Excel, který je schope apláovat ejkratší možou spojici pro 4 libovolých bodů s ávratem do bodu výchozího. Model, kterému jsem dal ozačeí Edita 3, řeší úlohu zámou pod termíem,,úloha obchodího cestujícího ovšem zobecěou pro libovolý počet míst až 4. Key words: Logistics, optimizatio of routes, mathematical models, liear programmig. Itroductio Today idustrial productio uses a lot of cooperative structures. Oe car cotais about differet compoets from may suppliers from differet cities ad coutries. This situatio puts high demads o areas related to orderig, shippig ad storage. Ad so started the disciplie logistics. Ay productio ca work whe logistics does t work properly. Trasport ad distributio of compoets ad fiished products is becomig a very importat part of the maagemet ad orgaizatio of productio. There s a kow problem i route plaig called,,travellig salesma problem. My paper describes a mathematical model solvig this problem for 4 poits. What s travellig salesma problem? There is a set of poits ad my objective is plaig the route amog these poits. The route must itersect all poits ad a fiish of the route is at the same poit like a start of the route. Well, for example a salesma goes from a tow umber oe, he comes to all other tows but to every tow comes just oly oce ad there is the fiish of his route back at the tow umber oe. Certaily the route must be the shortest. There is the problem to fid a optimum route amog all poits. This problem ofte arises i a umber of logistical situatios (for example trasport compoets to factories, trasportatio of goods to the shops, check braches, distributio of mail, but also supply workplaces i assembly hall or maufacturig plat etc.). 2. To solutio of,,travellig salesma problem 2.. To describe situatio with mathematical form Solutio of,,travellig salesma problem is more complicated. There is a lot of combiatios of routes.

2 Picture : The umber of combiatios c the umber of combiatios the umber of poits (tows, statios, workplaces) c = (-) (-2) (-3) = (-)! Formula : The umber of combiatios Whe the umber of poits grows, the umber of combiatios icreases very rapidly. How to fid the shortest (optimum) route? The problem will have to be trasformed to mathematical form ad the it will be mathematical solved. The result is made up values of variables. Each variable symbolizes way betwee two poits. Whe is value zero, there is ot way ad whe is value oe there is way. Picture 2: Trasformatio usig mathematical variables

3 The solutio cotais four mathematical formulas. I m describig the priciple usig for 4 poits. F = i= j= c ij x ij = mi. F = 0 x x x x _ x _ + 88 x _ x _ x _4 + 0 x x x x 2_ x 2_ + 00 x 2_ x 2_3 + 0 x 2_4.. x 3_ +... x 4_ = mi. Formula 2: Total legth of the route i= xij =, x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x _0 + x _ + x _2 + x _3 + x _4 = x 2 + x 23 + x 24 + x 25 + x 26 + x 27 + x 28 + x 29 + x 2_0 + x 2_ + x 2_2 + x 2_3 + x 2_4 =..... x 4_ + x 4 _ 2 + x 4_3 + x 4_4 + + x 4_8 + x 4_9 + x 4 _ 0 + x 4_ + x 4_2 + x 4_3 = Formula 3: Coditio Every poit may be leaved just oce j= x ij =, x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 0_ + x _ + x 2_ + x 3_ + x 4_ =..... i =,2,... j =,2,... x _4 + x 2_4 + x 3_4 + x 4_4 + x 5_4 + + x 8_4 + x 9_4 + x 0_4 + x _4 + x 2_4 + x 3_4 = Formula 4: Coditio Every poit may be visited just oce u u i j i =,2,... j =,2,... + x u 2 u 3 + 4x 23 3 u 3 u 2 + 0x 32 3,..... u 3 u 4 + 4x 3_4 3 u 4 u 3 + 4x 4_3 3 Formula 5: Coditio Route forms cotiuous circuit ij

4 xij = 0 ; Formula 6: Coditio variable is oly oe or zero I m lookig for values of variables to total legth of the route will be miimum (Formula 2) ad simultaeously coditios 3, 4, 5 ad 6 will have to valid. For 4 poits there are 95 variables, 84 coditios ad combiatios. This problem is with aother ad aother poit more complicated. I have bee described to trasformatio to mathematical otatio, but I eed to get a result ad the optimal route To calculatio mathematical otatio I m calculatig values of variables ad I m usig a special mathematical tool called,,liear Programmig ad because I eed oly values oe or zero I will use modificatios of Liear Programmig ad it is,,biary Liear Programmig. Picture 3: Usig Biary Liear Programmig i Excel This tool is i program Excel its ame is,,řešitel. Mathematical form will be writte to form for Excel ad Excel will give solutio. Picture 4: The procedure of fidig optimal route Excel ca calculatio Liear Programmig oly for maximum 200 ukow variables. It is very importat, because I sad i my paper, that: for 4 poits there are 95 variables. 4 poits is the upper limit for solutio i Excel. The procedure of fidig the optimal route is so difficult ad legthy. I m describig i the ext part of this paper my model for easy solutio this problem (travellig salesma problem) with maximum 4 poits. I amed this model EDITA 3.

5 3. My model for fidig optimal routes amed EDITA 3 by me The priciple, which I described i the first part of my paper, was used i this model. All calculatios were programmed. Operatio of the program is very simple just iput distaces ad the ames of the poits (for example tows) ad the ru the tool,,řešitel. The model automatically fids the result, which is usig fuctios of Excel trasformed from values of,,0 ad o the listig of poits (tows) formig the optimal route. This model ca work quite geerally with a umber of poits to 4. There is a differece to covetioal formulas. User does ot eed a static model, but the model must be able to work with chagig umber of cities. Picture 5: Iput date to model EDITA 3 (part of picture) Picture 6: Output date from model EDITA 3 (part of picture)

6 4. Coclusio Travellig salesma problem is a kow task i logistics. There are a lot of books with kow formulas but my model is origial, because it is able to trasform these static formulas i a dyamic workig model i the rage to 4 poits. Typical applicatios for this model is fidig the shortest route, if specified distaces betwee the poits. You ca also specify the times of joureys betwee the poits ad the result would be ot the shortest, but the fastest route. Aother task for future work is to look for ways to icrease the maximum umber of poits. Literature:. Fábry, J.: Matematické modelováí. Praha: VŠE, s. ISBN Jabloský, J.: Programy pro matematické modelováí. Praha: VŠE, s. ISBN Kožíšek, J., Stieberová B.: Statistická a rozhodovací aalýza. Praha: ČVUT, s. ISBN Schels, I.: Excel 2007 vzorce a fukce. Praha: Grada Publishig, s. ISBN Štůsek, J.: Řízeí provozu v logistických řetězcích. Praha: C. H. Beck, s. ISBN