Transportation problems Operational Research Level 4

1. Transportation problemsOperational Research Level 4 Prepared by T.M.J.A.Cooray Department of Mathematics MA 4020-Transportation problems

2. Introduction • Transportation problem is a special kind of LP problem in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the source and the destination respectively, such that the total cost of transportation is minimized. MA 4020-Transportation problems

3. Sources factories, finished goods warehouses , raw materials ware houses, suppliers etc. Destinations Markets Finished goods ware house raw materials ware houses, factories, Examples: MA 4020-Transportation problems

4. A schematic representation of a transportation problem is shown below D1 a1 a2 ai am S1 b1 b2 bj bn D2 Si Dj Dn Sm MA 4020-Transportation problems

5. m- number of sources • n- number of destinations • ai- supply at source I • bj – demand at destination j cij – cost of transportation per unit from source i to destination j Xij – number of units to be transported from the source i to destination j MA 4020-Transportation problems

6. 1 2 j n Supply a1 a2 ai am 1 2 i m S O U R C E i • Destination j MA 4020-Transportation problems Demand b1 b2 bj bn

7. Transportation problem: represented as a LP model MA 4020-Transportation problems

8. The ideal situation is shown below.,with equalities instead of inequalities. There are “mn” unknown variables and m+n-1 independent equations. MA 4020-Transportation problems

9. When solving the transportation problem ,the number of possible routes should be  m+n-1. If it is <m+n-1, it is called a degenerate solution. In such a case evaluation of the solution will not be possible. In order to evaluate the cells /routes (using the u-v method or the stepping stone method ) we need to imagine/introduce some used cells/routes carrying / transporting a very small quantity, say . That cell should be selected at the correct place. MA 4020-Transportation problems

10. Example: Consider a transportation problem involving 3 sources and 3 destinations. MA 4020-Transportation problems

11. Types of transportation problems • Balanced transportation problems • Unbalanced transportation problems Include a dummy source or a dummy destination having a supply “d” or demand “d” to convert it to a balanced transportation problem. Where d= MA 4020-Transportation problems

12. Example Plant W A 1 R E 2 H O 3 U S 4 E Supply MA 4020-Transportation problems

13. Solution of transportation problems • Two phases: • First phase: • Find an initial feasible solution • 2nd phase: • Check for optimality and improve the solution MA 4020-Transportation problems

14. Find an initial feasible solution • North west corner method • Least cost method • Vogel’s approximation method MA 4020-Transportation problems

15. Checking for optimality • U-V method • Stepping-Stone method MA 4020-Transportation problems

16. Example-( having a degenerate solution)Introduce  to for phase 2.. Destinations Sources S1 S2 S3 S4 Demand MA 4020-Transportation problems

17. Transshipment models. • In transportation problems ,shipments are sent directly from a particular source to a particular destination to minimize the total cost of shipments. • It is sometimes economical if the shipment passes through some transient nodes in between the sources and destinations. • In transshipment models it is possible for a shipment to pass through one or more intermediate nodes before it reaches its destination. MA 4020-Transportation problems

18. Transshipment problem with sources and destinations acting as transient nodes • Number of starting nodes as well as the number of ending nodes is the sum of number of sources and the number of destinations of the original problem. • Let B= • be the buffer stock and it is added to all the starting nodes and all the ending nodes. MA 4020-Transportation problems

19. a1+B aj+B am+B B B S1 B B B b1+B bn+B S1 .. .. … … Sj Sj Sm Sm D1 D1 Dn Dn MA 4020-Transportation problems

20. Destinations D1,D2,….Dn are included as additional starting nodes mainly to act as transient nodes.they don’t have any original supply and the supply of these nodes should be at least B. • The sources S1,S2,….Sm are included as additional ending nodes mainly to act as transient nodes.these nodes are not having any original demand.But each of these transient nodes is assigned with B units as the demand value. MA 4020-Transportation problems

21. We need to know the transshipment cost between the sources ,between the destinations and between sources and destinations . MA 4020-Transportation problems

22. Example • Supplies at the sources are 100,200,150 and 350 and Demand at the destinations are 350 and 450 respectively. MA 4020-Transportation problems

23. Same algorithms can be used to solve this transshipment problem. MA 4020-Transportation problems

24. Transportation problem with some transient nodes between sources and destination. • Consider the case where the shipping items are first sent to intermediate finished goods ware houses from the supply points/factories and then to the destinations. • To solve these problems the capacity at each transient node is made equal to B. • Where B = MA 4020-Transportation problems

25. Example • Multi plant organization has 3 plants and three market places. • The goods from the plants are sent to market places through two intermediate finished goods warehouses. • Cost of transportation per unit between plants and warehouses and warehouses to market places and also supply values of plants and demand values of the markets are shown in the table. MA 4020-Transportation problems

26. 900 900 900 900 900 Solution of the problem is same as Ordinary transportation Problems. MA 4020-Transportation problems

27. MA 4020-Transportation problems