1. Distribution Model Meaning Types Transportation Model Assignment Model

2. Distribution Models Distribution problems are a special type of linear programming problem. Two types of Model. They are: i.Transportation Problems: Deals with shipment from number of sources to number of destinations. – Typically each source is supply limited and each destination has a known demand ii.Assignment Problems: Deals with finding the best one to one match for each of a given no of candidates. As for example: – Assigning worker to machine – Assigning teacher to classes Objective is to get maximum reward or minimum cost.

3. Transportation Model Wheat is harvested in Midwest and stored in grain elevators in three diff cities-Kansas Omaha and des moines. Three grain elevators supply three flour mills located in Chicago, St. Louis and Cincinnati. Each grain elevator is able to supply the following no of tons of wheat to mills. ElevatorsSupply 1Kansas150 2Omaha175 3Des Moines275 Total600 tons Each Mill demand the following no of tons MillDemand AChicago200 BSt. Louis100 CCincinnati300 Total600 tons The cost of transporting one ton of wheat from each grain elevator to each mill are Grain elevators A. Chicago B. St. Louis C. Cincinnati 1.Kansas6810 2. Omaha711 3. Des Moines 4512 The problem is to how many tons of wheat to transport from each grain elevators to each mill on a monthly basis in order to minimize total cost of transportation

4. Solution of Transportation Model Several methods are: i.The northwest corner methods ii.Minimum cell cost method iii.Vogel’s approximation method(VAM) iv.The stepping- stone solution method

5. Solution of Transportation Model Transportation models are solved manually with in the context of a tableau as in the simplex method. From ToABCSupply 1 6810 150 2 711 175 3 4512 275 Demand200100300600

6. Solution of Transportation Model The northwest corner method: Here the largest possible allocation is made to the cell in the upper left hand corner of the tableau, followed by allocation to adjacent feasible cells. From ToABCSupply 1 6810 150 2 711 175 3 4512 275 Demand200100300600

7. Solution of Transportation Model The northwest corner method: Here the largest possible allocation is made to the cell in the upper left hand corner of the tableau, followed by allocation to adjacent feasible cells. i.Start with the northwest corner, allocate the smaller amount of either the row supply or column demand ii.Subtract from the row supply and the column demand the amount allocated iii.If the column demand is zero move to the cell next on the right if the row supply is zero move down to the cell in the next row. If both are zero move first to the next cell on the right then down on cell

8. Solution of Transportation Model The northwest corner method: Here the largest possible allocation is made to the cell in the upper left hand corner of the tableau, followed by allocation to adjacent feasible cells. From ToABCSupply 1 150 6810 150 2 50 7100 1125 11 175 3 45275 12 275 Demand200100 300600

9. Solution of Transportation Model The northwest corner method: Here the largest possible allocation is made to the cell in the upper left hand corner of the tableau, followed by allocation to adjacent feasible cells. Total cost =150*6+50*7+100*11+25*11+275*12=5925 From ToABCSupply 1 150 6810 150 2 50 7 100 1125 11 175 3 45275 12 275 Demand200100 300600

10. Solution of Transportation Problems (least cost Method) Logic is to allocate to the cells with the lowest cost The next allocation is made the cell that has the minimum cost Repeat this way until all rim requirements have been made.

11. Solution of Transportation Model (least cost method) From ToABCSupply 1 625 8125 10 150 2 711 175 11 175 3 200 475 512 275 Demand200100300600 Total cost =4550

12. Solution of Transportation Problems (least cost Method) Logic is to allocate to the cells with the lowest cost The next allocation is made the cell that has the minimum cost Repeat this way until all rim requirements have been made.

13. Solution of Transportation Problems (least cost Method) Logic is to allocate to the cells with the lowest cost The next allocation is made the cell that has the minimum cost Repeat this way until all rim requirements have been made.

14. Solution of Transportation Problems (least cost Method) Logic is to allocate to the cells with the lowest cost The next allocation is made the cell that has the minimum cost Repeat this way until all rim requirements have been made.

15. Solution of Transportation Problems (least cost Method) Logic is to allocate to the cells with the lowest cost The next allocation is made the cell that has the minimum cost Repeat this way until all rim requirements have been made.

16. Solution of Transportation Model Balance the Table Transportation solution technique requires that the problem be balanced ie total supply must equal total demand. Two causes of imbalance are excess supply and excess demand. i.Excess supply: If it is present in any table first balance it by adding a dummy column (dummy destination). ii.Excess demand: When total demand exceeds total supply a dummy source row is added to balance the table. In both case the shipment costs are set to zero

17. Solution of Transportation Model Balance the Table i.Excess supply: If it is present in any table first balance it by adding a dummy column (dummy destination). Source/ destinationRSTSupply A123200 B415100 Demand8012060260/300 Source/desti nation RSTDummySupply A1230200 B4150100 Demand801206040(excess supply) 300/300

18. Solution of Transportation Model Balance the Table i.Excess demand: When total demand exceeds total supply a dummy source row is added to balance the table. Source/ destinationRSTSupply A123100 B415110 Demand8012060260/210 Source/destinationRSTSupply A123100 B415110 Dummy row00050 Demand8012060260/260

19. Solution of Transportation Model Balance the Table: Northwest corner solution i.Excess demand: When total demand exceeds total supply a dummy source row is added to balance the table. Source/ destinationRSTSupply A123100 B415110 Demand8012060260/210 Source/destinationRSTSupply A80 120 23100 B4100 110 5110 Dummy row00 50 050 Demand8012060260/260

20. Assignment Model The Hungarian Method: The following algorithm applies the above theorem to a given n × n cost matrix to find an optimal assignment. Step 1. Subtract the smallest entry in each row from all the entries of its row. Step 2. Subtract the smallest entry in each column from all the entries of its column. Step 3. Draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used. Step 4. Test for Optimality: (i) If the minimum number of covering lines is n, an optimal assignment of zeros is possible and we are finished. (ii) If the minimum number of covering lines is less than n, an optimal assignment of zeros is not yet possible. In that case, proceed to Step 5. Step 5. Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to Step 3.

21. Example 1 You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are in Austin, TX; Boston, MA; and Chicago, IL. You want them to fly to three other cities: Denver, ; Edmonton, and Fargo, ND. The table below shows the cost of airplane tickets in dollars between these cities. Where should you send each of your salespeople in order to minimize airfare? From/ToDenverEdmontonFargo Austin250400350 Boston400600350 Chicago200400250

22. Example 1 Step 1. Subtract 250 from Row 1, 350 from Row 2, and 200 from Row 3. From/ToDenverEdmontonFargo Austin250400350 Boston400600350 Chicago200400250 From/ToDenverEdmontonFargo Austin0150100 Boston502500 Chicago020050

23. Example 1 Step 2. Subtract 0 from Column 1, 150 from Column 2, and 0 from Column 3. From/ToDenverEdmontonFargo Austin0150100 Boston502500 Chicago020050 From/ToDenverEdmontonFargo Austin00100 Boston501000 Chicago050

24. Example 1 Step 3. Cover all the zeros of the matrix with the minimum number of horizontal or vertical lines. From/ToDenverEdmontonFargo Austin00100 Boston501000 Chicago050 Step 4. Since the minimal number of lines is 3, an optimal assignment of zeros is possible and we are finished. AssignmentCost Austin -Edmonton400 Boston-Fargo350 Chicago-Denver200

25. Example 2 Machine/JobABCD 18524 2671110 33576 45 129 Determine the optimum assignment of jobs to machine for the Data

26. Example 3 Machine/JobABCD 112161410 298137 31512911 Table contains information on cost to run 3 jobs on four machines.Determine an assignment plan that will minimize cost

27. Solution of Transportation Model

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