transporation problem - stepping stone method

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Transportation Problem - Optimal Solution using Stepping Stone Method

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transporation problem - stepping stone method

  1. 1. Transportation Problem- Stepping Stone Method -PAMANTASAN NG LUNGSOD NG MAYNILAGRADUATE SCHOOL OF ENGINEERINGGEM 805 – OPTIMIZATION TECHNIQUES
  2. 2. Stepping Stone Method>>> This is a one of the methods used to determine optimality ofan initial basic feasible solution (i.e. Northwest Corner Rule, LeastCost or Vogel’s Approximation)>>> The method is derived from the analogy of crossing a pondusing stepping stones. This means that the entire transportationtable is assumed to be a pond and the occupied cells are thestones needed to make certain movements within the pond.
  3. 3. Optimum Solution:Stepping-Stone Method1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSZ = 4x10+6x30+6x50+7x10+5x10+8x40 = 960Transportation Table
  4. 4. 1. Starting at an unused/empty cell, trace a closed path or loop backto the original cell via cells that are currently being used and/oroccupied.Note: A closed path or loop is a sequence of cells in thetransportation table such that the first cell is unused/emptyand all the other cells are used/occupied with the followingconditions:a. Each pair of consecutive used/occupied cells lies in either thesame row or columnb. No three consecutive used/occupied cells lie in the same rowor columnc. The first and last cells of a sequence lies in the same row orcolumnd. No cell appears more than once in a sequence (i.e. noduplication)e. Only horizontal and vertical moves allowed and can onlychange directions at used/occupied cellsOptimum Solution:Stepping-Stone Method
  5. 5. Example:1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodA3->B3->B4->C4->C1->A1->A3At Cell A3,
  6. 6. Example:1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodAt Cell A4, A4->C4->C1->A1->A4
  7. 7. 1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150Optimum Solution:Stepping-Stone MethodB1->B4->C4->C1->B1SOURCESDESTINATIONSExample: At Cell B1,
  8. 8. Example:1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodAt Cell B2, B2->B4->C4->C1->A1->A2->B2
  9. 9. Example:1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodAt Cell C2, C2->C1->A1->A2->C2
  10. 10. Example:1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodAt Cell C3, C3->B3->B4->C4->C3
  11. 11. 2. For every traced path or loop, begin with a plus (+) sign at thestarting unused cell and alternately place a minus (-) and plus (+)sign at each used cell1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONS---+++Example:Optimum Solution:Stepping-Stone MethodAt Cell A3, A3->B3->B4->C4->C1->A1->A3
  12. 12. 3. Calculate an Improvement Index by first adding the unit-costfigures found in each cell containing a plus sign and subtractingthe unit costs in each square containing a minus sign.1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONS---+++Example: At Cell A3, A3->B3->B4->C4->C1->A1->A3Optimum Solution:Stepping-Stone MethodIA3 = 2-8-6 +7+5-4=8
  13. 13. Optimum Solution:Stepping-Stone MethodIteration #1 - Computing for the Improvement Index:At A3, A3->B3->B4->C4->C1->A1; IA3 = +8-6+7-8+5-4 = 2At A4, A4->C4->C1->A1; IA4 = +8-8+5-4 = 1At B1, B1->B4->C4->C1; IB1 = +6-7-8-5 = 2At B2, B2->B4->C4->C1->A1->A2; IB2 = +8-7+8-5+4-6 = 2At C2, Loop C2->C1->A1->A2; IC2 = +7-5+4-6 = 0At C3, C3->B3->B4->C4; IC3 = +6-6+7-8 = -14. If all indices calculated are greater than or equal to zero, then,an optimal solution had been reached. If not, select thepath/loop that has the most negative value and use this tofurther improve the solution.Note: Should there be two or more “most” negative values,select arbitrarily.
  14. 14. Example: At Cell C3, C3->B3->B4->C41 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone Method++--IC3 = +6-6+7-8 = -1
  15. 15. To further improve the current solution, select the “smallest” number foundin the path/loop C3->B3->B4->C4 containing minus(-) signs. This number isadded to all cells on the closed path/loop with plus(+) signs and subtractedfrom all cells on the path assigned with minus(-) signs.Optimum Solution:Stepping-Stone Method1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76050 10C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONS++--4050 - 40 10 + 4040 - 40
  16. 16. 5. Then, we have a new basic feasible solution…1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76010 50C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone Method…and repeat steps 1 though 4 to calculate an Improvement Index forall unused squares in order to test whether an optimal solution hasbeen reached.
  17. 17. Optimum Solution:Stepping-Stone MethodIteration #2 - Computing for the Improvement Index:At A3, A3->C3->C1->A1; IA3 = +8-6+5-4 = 3At A4, A4->B4->B3->C3->C1->A1; IA4 = +8-7+6-6+5-4 = 2At B1, B1->B3->C3->C1; IB1 = +6-6+6-5 = 1At B2, B2->B3->C3->C1->A1->A2; IB2 = +8-6+6-5+4-6 = 1At C2, C2->C1->A1->A2; IC2 = +7-5+4-6 = 0At C4, C3->B3->B4; IC3 = +8-6+6-7 = 1Since the results of all indices calculated are greater than or equal tozero, then, an optimal solution had been reached.
  18. 18. …and computing the objective function Z:1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76010 50C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodZ = 4x10+6x30+6x10+7x50+5x10+6x40 = 920
  19. 19. In Iteration #2 :At A3, A3->C3->C1->A1; IA3 = +8-6+5-4 = 3At A4, A4->B4->B3->C3->C1->A1; IA4 = +8-7+6-6+5-4 = 2At B1, B1->B3->C3->C1; IB1 = +6-6+6-5 = 1At B2, B2->B3->C3->C1->A1->A2; IB2 = +8-6+6-5+4-6 = 1At C2, C2->C1->A1->A2; IC2 = +7-5+4-6 = 0At C4, C3->B3->B4; IC3 = +8-6+6-7 = 1Optimum Solution:Stepping-Stone MethodHowever, in checking the calculation in Iteration #2, there is animprovement index equal to zero. This means that there is anALTERNATE optimum solution:
  20. 20. To calculate for the alternate optimum solution, again select the “smallest”number found in this path/loop containing minus(-) signs. This number isadded to all cells on the closed path/loop with plus(+) signs and subtractedfrom all cells on the path assigned with minus(-) signs.Optimum Solution:Stepping-Stone Method1 2 3 4 SUPPLYA4 6 8 84010 30B6 8 6 76010 50C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONS++ -- 1030 - 1010 + 1010 - 10Hence, at C2->C1->A1->A2,
  21. 21. Then the alternate optimum solution with objective function Z:1 2 3 4 SUPPLYA4 6 8 84020 20B6 8 6 76010 50C5 7 6 85010 40DEMAND 20 30 50 50 150SOURCESDESTINATIONSOptimum Solution:Stepping-Stone MethodZ = 4x20+6x20+6x10+7x50+7x10+6x40 = 920
  22. 22. 1 2 3 4 SUPPLYA4 6 8 84020 20B6 8 6 76010 50C5 7 6 85050DEMAND 20 30 50 50 150SOURCESDESTINATIONSWhen the number of empty/occupied cells in any solution (eitherinitial or later) of the transportation table is not equal to the numberof rows plus the number of columns minus 1 (i.e. m+n-1) thesolution is called DEGENERATEOptimum Solution:Stepping-Stone MethodExample: m + n -1 = 3 + 4 -1 = 6DEGENERACY
  23. 23. DEGENERACYTo handle degenerate problems, artificially create an occupied cell byplacing a zero (representing a fake shipment) in one of the unusedcells. Treating this cell as if it were occupied, it must be chosen in sucha position as to allow all stepping-stone paths to be traced. Then, allstepping-stone paths can be closed and improvement indicescomputed.Optimum Solution:Stepping-Stone Method1 2 3 4 SUPPLYA4 6 8 84020 20B6 8 6 76010 50C5 7 6 85050DEMAND 20 30 50 50 150SOURCES0Example: DESTINATIONS
  24. 24. QUESTIONS?Optimum Solution:Stepping-Stone Method
  25. 25. DIOS MABALOS PO!Cam on !Shukriya !ありがとうございます!Thank you!Merci!Gracias!Obrigado!謝謝!
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