3. Transportation Problem (Part 2)

Transcription

1 3. Transportation Problem (Part 2) 3.6 Test IBFS for optimal solution or Examining the Initial Basic Feasible Solution for Non- Degeneracy. 3.7 Transportation Algorithm for Minimization Problem (MODI Method) 3.8 Unbalanced Transportation Problem 3.9 Degeneracy in Transportation Problem 3.10 Maximization in Transportation problem 3.11 Questions of Transportation Problem

2 3.6 Test IBFS for optimal solution or Examining the Initial Basic Feasible Solution for Non-Degeneracy After computing an IBFS, we must check whether the solution so obtained is optimal or not. IBFS obtained by any of the three methods should be non- degenerate basic feasible solution. A Basic Feasible Solution is called non-degenerate if number of positive allocations x ij = m+n-1 and these m+n-1 allocations must be in independent positions Independent Position means set of positive allocations x ij which do not form a closed loop. Closed loop: a closed loop is a collection of horizontal and vertical line in which each corner of loop should be in occupied cell. Independent Positions Non-Independent Positions

3 3.7 Transportation Algorithm for Minimization Problem (MODI Method) Step 1 Construct the transportation table entering the origin capacities (supply) a i, the destination requirement (demand) b j and the cost c ij. Step 2 Find an initial basic feasible solution by Vogel s method or by any of the given method. Step 3 Test for optimality: If number of positive independent allocations x ij = m+n-1 Then IBFS is non-degenerate and optimality test can be performed. If number of positive independent allocations x ij < m+n-1 Then IBFS is degenerate and optimality test cannot be performed. And first we have to convert degenerate solution into non degenerate. Step 4 Apply MODI Method 1. Set up cost matrix for allocated (occupied) cell only. 2. Enter the set of number u i for row and v j for column such that for occupied cell c ij, = u i + v j. 3. Set any u i or v j = 0 which have maximum allocation to calculate the values of u i, v j. 4. Compute u i + v j for unoccupied cells. 5. Compute the opportunity cost d ij = c ij ( u i + v j ) for unoccupied cells. Step 5 Apply optimality test by examining the sign of each d ij If all d ij 0, the current initial basic feasible solution is optimal. If at least one d ij < 0, the current initial basic feasible solution is not optimal and we have to improve the solution.

4 Step 6 Iterative towards an optimal solution: Identify the most negative cell in d ij matrix and called this cell as identified cell. (The most negative value is the rate by which total transportation cost is reduced if one unit is transported to this cell) Write IBFS again which needs to be improved. Marked ( ) the identified cell and make a closed loop starting and ending in the identified cell. All the corner of the loop should be on occupied cell except identified cell. Mark the identified cell as +ive and other corner of the loop with ive, +ive, ive sign alternatively. Find minimum allocated value on the loop where we have ive sign. Add these minimum allocations in the loop where we have +ive sign and subtract where we have ive sign. ( by these most negative cell gets an allocation and minimum allocated cell with ive sign gets zero allocation) By doing all the above steps we get second basic feasible solution. Step 7 Now, return to step 4 and repeat the process until an optimal solution is obtained. Find optimal solution of given transportation Problrm. Obtain initial basic feasible solution by VAM (Vogel s method). Warehouse Factory W 1 W 2 W 3 W 4 Capacity F F F Requirement

5 Initial basic feasible solution by using Vogel s Approximation Method

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13 3.8 Unbalanced Transportation Problem Total Supply Total Demand cases So, Problem is unbalanced. Now to make problem balanced there arise two Case I: If Total Supply > Total Demand then We add a dummy demand centre or column Unit cost of dummy column is zero and demand for dummy column = Total Supply Total Demand Case II: If Total Supply < Total Demand then We add a dummy supply centre or row Unit cost of dummy row is zero and demand for dummy column = Total Demand Total Supply

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20 3.9 Degeneracy in Transportation Problem If the number of positive allocations is less than m+n-1 then it is called as Degenerate Basic Feasible Solution. Resolution of Degeneracy : To resolve Degeneracy, we allocate a small positive number epsilon to the unoccupied cell which have minimum cost and should be on independent position. The cell where we allocate epsilon is treated like other occupied cell and degeneracy is removed. So we have number of positive allocations x ij = m+n-1 so, solution is Non - Degenerate Basic Feasible Solution.

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25 3.10 Maximization Transportation Problem To convert profit matrix into loss matrix first check problem is balanced or not. If problem is not balanced then make it balanced. Now convert profit matrix into loss matrix subtract all elements of profit matrix from highest profit.

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27 3.11 Questions on Transportation Problem 1. What is Transportation problem? 2. Write mathematical model of Transportation Problem? 3. Define following terms in context to Transportation Problem : feasible solution, basic feasible solution, non-degenerate basic feasible solution, degenerate basic feasible solution, optimal solution? 4. Define loop in a Transportation table. What role do they play? 5. Write steps to find the initial basic feasible solution by North-West corner rule. 6. Write steps to find the initial basic feasible solution by Least Cost method. 7. Write steps to find the initial basic feasible solution by Vogel s Approximation Method. 8. What is degeneracy in transportation problem? How it is resolved. 9. Explain briefly about Unbalanced Transportation Problem. 10. What is Unbalanced Transportation Problem? How do you start in this case? 11. A company is sending Rs. 1,000 on transportation of its units from three plants to four distribution centers. The supply and demand of units with unit cost of transportation are given as Distribution center D 1 D 2 D 3 D 4 Capacity(tonnes) Plant P P P emand Find initial basic feasible solution by N.W.C.R. and by Vogel s method. Find Optimal solution also. Answer: NWCR = 894, VAM = Solve the following Transportation problem and find the optimal solution: To W 1 W 2 W 3 W4 Supply From D D D Demand Answer : VAM = 796 (it is optimal cost after applying MODI method one time).

28 13. Solve the following transportation problem for minimization. D 1 D 2 D 3 D 4 Supply S S S Demand ( Hint : Make problem balanced first then find IBFS by VAM) Answer: VAM = 110 ( apply MODI method solution is not optimal, improve solution then cost by VAM = 107 again apply MODI method and solution is optimal. 14. Solve the transportation problem for which the cost, origin availabities and destination requirement are given below D 1 D 2 D 3 D 4 Supply Demand Hint : problem is balanced, Find IBFS by VAM Answer : VAM = 848, IBFS is degenerate, make it non degenerate and then apply MODI method one time to get Optimal solution. 15. Solve the Transportation Problem for Maximization Factory Depot S 1 S 2 S 3 Availability F F F F Requirement Hint : see maximization Transportation solved problem above. Answer : VAM = 480 (maximum Profit) and apply MODI Method once to get optimal solution.