graph theory

Graph Theory Project, Kutztown University Mathematics Department. ′Approveof Dr. Amy Lu′

Application of Graph Theory with thecombination of Operation Research to find

Optimal pathsv. 1.0.0

Toan Nguyen and Grace [email protected] and [email protected]

April 27, 2016

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Content

IntroductionOverview of Graph Theory and Operation Research

Kruskal’s AlgorithmDefinition of Kruskal’s Algorithm

Northwest CornerDefinition of Northwest CornerRules by Joshi

Conclusion

References

Toan Nguyen and Grace Liu | Toan Nguyen and Grace Liu

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IntroductionGraph Theory and Operation Research

I Graph theory is a wise world known applications in OperationResearch. By using a graph to define as a finite number of pointswhere those number of points are represented nodes or verticesof the graph. (See below figure 1) whereas nodes or vertices (inblue), and (in green) are represented edges of a graph. In thispaper for a given graph using (real world) problem to find aminimum cost and shortest path from one point to another.


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I Different techniques have been use widely around the world,especially factory/logistic companies. They frequently try to finda shortest path to ship the products from one place to another,within a certain constraint, as well as without exceeding thebudget, before the products get ship to consumers. UsingKruskal’s Algorithm and Dijkstra’s Algorithm to help Japancompanies to find a shortest path to ship their products to theU.S, as well as to satisfy demand and supply availability, fromJapan to the U.S is 40 demand and 40 supply, vice versa. (SeeFigure 2 and 3)


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Kruskal’s Algorithm

I There are many different ways to go from one node to another(See figure 2), according to Rame Likaj. The object of thisproject is to define a shortest path from Japan shipping productsto the U.S with a minimum transportation costs, will beintroduced later in this paper. Our objective at this point is todefine how to get from point A (Japan) to point B (the U.S) byusing Kruskal Algorithm.


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I By Kruskal Algorithm’s on defining the shortest path, we will startwith the lowest cost to the one with higher costs without creatingcycles (See figure 3)


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I From figure 2 and 3 we finally found one path that is consideredas the minimum spanning tree path from nodes A to the U.S,according to Rame. Further explanation:

I Further explanation:(Red line) first option starting from: 7 + 5 + 2 + 4 + 3 + 4 + 2 =27 units cost.(Green line) second option starting from:13 + 4 + 3 + 8 + 3 + 6 + 4 + 8 + 10 + 4 + 11 + 5 + 7 = 86 unitscost (which also consider as the most expensive path)From both options that we obtained, we can conclude that thefirst option is represented the minimum cost path from Japan tothe U.S.


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Northwest Corner

I Using the Northwest Corner method to help to define thedemand and supply to find minimum transportation cost


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Northwest CornerFollowing the bottom rules, by Joshi

I 1. Starting from the northwest corner of the transportationtableau, allocate as much quantity as possible to cell (1,1) fromOrigin 1 to Destination 1, within the supply constraint of source 1and the demand constraint of destination 1.

I 2. The first allocation will satisfy either the supply capacity ofSource 1 or the destination requirement of Destination 1.a. If the demand requirement for destination 1 is satisfied but thesupply capacity for Source 1 is not exhausted, move on to cell(1,2) for next allocation.b. If the demand requirement for destination 1 is not satisfied butthe supply capacity for Source 1 is exhausted, move to cell (2,1)c. If the demand requirement for Destination 1 is satisfied andthe supply capacity for Source 1 is also exhausted, move on tocell (2,2).

I 3. Continue the allocation in the same manner toward thesoutheast corner of the transportation tableau until the supplycapacities of all sources are exhausted and the demands of alldestinations are satisfied.


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Northwest Corner MethodExample


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Once the feasible solution been defined in order word, once thesupply and demand both have been satisfied, we now then calculatethe total cost by multiplying each xij in an occupied cell with thecorresponding cij as adding as follows:[(10)X5] + [(10)X15] + [(30)X15] = 650 (equal to the totaltransportation cost)Applying the same techniques to find the transportation cost (toprove) our path from Japan shipping to the U.S that found earlier isthe lowest total transportation cost.


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Applying the same technique

Note: Demands and Supply both are randomly number assignedfrom the original cost


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Once the feasible solution been defined in order word, once thesupply and demand both have been satisfied, we now then calculatethe total cost by multiplying each xij in an occupied cell with thecorresponding cij as adding as follows:[(10)X7] + [(5)X5] + [(5)X2] + [(4)X5] + [(5)X2] = 135 (equal to thetotal transportation cost)


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Using the same techniques for the most expensive path (86) thatfound from Kruskal’s Algorithm


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Once the feasible solution been defined in order word, once thesupply and demand both have been satisfied, we now then calculatethe total cost by multiplying each xij in an occupied cell with thecorresponding cij as adding as follows:[(5)X13] + [(4)X4] + [(3)X3] + [(3)X8] + [(2)X4] + [(2)X10] +[(4)X8] + [(2)X4] = 182 (equal to the total transportation cost)


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Conclusion

Using Kruskal’s algorithm and with the assisting of the northwestcorner method to find the most optimal path in both lowest cost andexpensive cost. We come up with the conclusion to prove that thelowest cost come out to be 135 of total transportation cost, while themost expensive cost come out to be 182 of total transportation cost.In other word, by using three techniques to solve this model and toanalyze the most effective way to minimize shipment costs of thetransportation problem come out to be effectively.


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References

[1] “Transportation Problem." MathStat Nikhil M Joshi. N.p., 28 Mar.2009. Web. 2 Apr. 2016.[2] "Kruskal’s Algorithm." - Lecture by Rashid Bin Muhammad, PhD.N.p., n.d. Web. 10 Apr. 2016.[3] Likaj, Ramë. "Application of Graph Theory to Find Optimal Pathsfor the Transportation Problem." 15th IFAC Workshop on InternationalStability, Technology, and Culture, 2013 (2013): n. pag. Web. 2 Apr.2016.[4] "Math Insight." N.p., n.d. Web. 29 Mar. 2016.[5] "Transportation Problem." MathStat Nikhil M Joshi. N.p., 28 Mar.2009. Web. 2 Apr. 2016.[6] "Transportation Problem." Transportation Problem. N.p., n.d. Web.10 Apr. 2016.


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